Symbolic Integration¶
- class sage.symbolic.integration.integral.DefiniteIntegral[source]¶
Bases:
BuiltinFunctionThe symbolic function representing a definite integral.
EXAMPLES:
sage: from sage.symbolic.integration.integral import definite_integral sage: definite_integral(sin(x), x, 0, pi) # needs sage.libs.maxima 2
>>> from sage.all import * >>> from sage.symbolic.integration.integral import definite_integral >>> definite_integral(sin(x), x, Integer(0), pi) # needs sage.libs.maxima 2
from sage.symbolic.integration.integral import definite_integral definite_integral(sin(x), x, 0, pi) # needs sage.libs.maxima
- class sage.symbolic.integration.integral.IndefiniteIntegral[source]¶
Bases:
BuiltinFunctionClass to represent an indefinite integral.
EXAMPLES:
sage: # needs sage.libs.maxima 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
>>> from sage.all import * >>> # needs sage.libs.maxima >>> 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
# needs sage.libs.maxima 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]¶
alias of
integrate().
- sage.symbolic.integration.integral.integrate(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
selfhas 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 integralb– (optional) upper endpoint of definite integralalgorithm– (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 libgiac'libgiac'– use libgiac (alias for'giac')
To prevent automatic evaluation, use the
holdargument.See also
To integrate a polynomial over a polytope, use the optional
latte_intpackagesage.geometry.polyhedron.base.Polyhedron_base.integrate().EXAMPLES:
sage: x = var('x') sage: h = sin(x)/(cos(x))^2 sage: h.integral(x) # needs sage.libs.maxima 1/cos(x)
>>> from sage.all import * >>> x = var('x') >>> h = sin(x)/(cos(x))**Integer(2) >>> h.integral(x) # needs sage.libs.maxima 1/cos(x)
x = var('x') h = sin(x)/(cos(x))^2 h.integral(x) # needs sage.libs.maximasage: f = x^2/(x+1)^3 sage: f.integral(x) # needs sage.libs.maxima 1/2*(4*x + 3)/(x^2 + 2*x + 1) + log(x + 1)
>>> from sage.all import * >>> f = x**Integer(2)/(x+Integer(1))**Integer(3) >>> f.integral(x) # needs sage.libs.maxima 1/2*(4*x + 3)/(x^2 + 2*x + 1) + log(x + 1)
f = x^2/(x+1)^3 f.integral(x) # needs sage.libs.maxima
sage: f = x*cos(x^2) sage: f.integral(x, 0, sqrt(pi)) # needs sage.libs.maxima 0 sage: f.integral(x, a=-pi, b=pi) # needs sage.libs.maxima 0
>>> from sage.all import * >>> f = x*cos(x**Integer(2)) >>> f.integral(x, Integer(0), sqrt(pi)) # needs sage.libs.maxima 0 >>> f.integral(x, a=-pi, b=pi) # needs sage.libs.maxima 0
f = x*cos(x^2) f.integral(x, 0, sqrt(pi)) # needs sage.libs.maxima f.integral(x, a=-pi, b=pi) # needs sage.libs.maxima
sage: f(x) = sin(x) sage: f.integral(x, 0, pi/2) # needs sage.libs.maxima 1
>>> from sage.all import * >>> __tmp__=var("x"); f = symbolic_expression(sin(x)).function(x) >>> f.integral(x, Integer(0), pi/Integer(2)) # needs sage.libs.maxima 1
f(x) = sin(x) f.integral(x, 0, pi/2) # needs sage.libs.maxima
The variable is required, but the endpoints are optional:
sage: # needs sage.libs.maxima 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)
>>> from sage.all import * >>> # needs sage.libs.maxima >>> 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)
# needs sage.libs.maxima 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
holdparameter it is possible to prevent automatic evaluation, which can then be evaluated viasimplify():sage: integral(x^2, x, 0, 3) # needs sage.libs.maxima 9 sage: a = integral(x^2, x, 0, 3, hold=True); a integrate(x^2, x, 0, 3) sage: a.simplify() # needs sage.libs.maxima 9
>>> from sage.all import * >>> integral(x**Integer(2), x, Integer(0), Integer(3)) # needs sage.libs.maxima 9 >>> a = integral(x**Integer(2), x, Integer(0), Integer(3), hold=True); a integrate(x^2, x, 0, 3) >>> a.simplify() # needs sage.libs.maxima 9
integral(x^2, x, 0, 3) # needs sage.libs.maxima a = integral(x^2, x, 0, 3, hold=True); a a.simplify() # needs sage.libs.maxima
Constraints are sometimes needed:
sage: # needs sage.libs.maxima 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()
>>> from sage.all import * >>> # needs sage.libs.maxima >>> 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()
# needs sage.libs.maxima 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: assume(n == -1) # needs sage.libs.maxima sage: integral(x^n, x) # needs sage.libs.maxima log(x)
>>> from sage.all import * >>> assume(n == -Integer(1)) # needs sage.libs.maxima >>> integral(x**n, x) # needs sage.libs.maxima log(x)
assume(n == -1) # needs sage.libs.maxima integral(x^n, x) # needs sage.libs.maxima
Note that an exception is raised when a definite integral is divergent:
sage: # needs sage.libs.maxima 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.
>>> from sage.all import * >>> # needs sage.libs.maxima >>> 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.
# needs sage.libs.maxima 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: integrate(1/x^3, x, 1, infinity) # needs sage.libs.maxima 1/2
>>> from sage.all import * >>> integrate(Integer(1)/x**Integer(3), x, Integer(1), infinity) # needs sage.libs.maxima 1/2
integrate(1/x^3, x, 1, infinity) # needs sage.libs.maxima
The examples in the Maxima documentation:
sage: # needs sage.libs.maxima 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)
>>> from sage.all import * >>> # needs sage.libs.maxima >>> 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)
# needs sage.libs.maxima 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: _ = 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)) # needs sage.libs.maxima 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(sqrt(-I)*x) - 2*sqrt(2)*imag_part(erf((-1)^(1/4)*x)) + 2*sqrt(2)*real_part(erf((-1)^(1/4)*x)))
>>> 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)) # needs sage.libs.maxima 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(sqrt(-I)*x) - 2*sqrt(2)*imag_part(erf((-1)^(1/4)*x)) + 2*sqrt(2)*real_part(erf((-1)^(1/4)*x)))
_ = 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)) # needs sage.libs.maximaAlternatively, just use algorithm=’mathematica_free’ to integrate via Mathematica over the internet (does NOT require a Mathematica license!):
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))
>>> 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))
_ = var('x, y, z') # optional - internet f = sin(x^2) + y^z # optional - internet f.integrate(x, algorithm='mathematica_free') # optional - internetWe can also use Sympy:
sage: integrate(x*sin(log(x)), x) # needs sage.libs.maxima -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) # needs sage.libs.maxima -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)))
>>> from sage.all import * >>> integrate(x*sin(log(x)), x) # needs sage.libs.maxima -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) # needs sage.libs.maxima -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)))
integrate(x*sin(log(x)), x) # needs sage.libs.maxima integrate(x*sin(log(x)), x, algorithm='sympy') # needs sympy _ = var('y, z') (x^y - z).integrate(y) # needs sage.libs.maxima (x^y - z).integrate(y, algorithm='sympy') # needs sympyWe integrate the above function in Maple now:
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)
>>> 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)
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: A = integral(1 / ((x-4) * (x^4+x+1)), x); A # needs sage.libs.maxima integrate(1/((x^4 + x + 1)*(x - 4)), x)
>>> from sage.all import * >>> A = integral(Integer(1) / ((x-Integer(4)) * (x**Integer(4)+x+Integer(1))), x); A # needs sage.libs.maxima integrate(1/((x^4 + x + 1)*(x - 4)), x)
A = integral(1 / ((x-4) * (x^4+x+1)), x); A # needs sage.libs.maxima
Sometimes, in this situation, using the algorithm “maxima” gives instead a partially integrated answer:
sage: integral(1/(x**7-1), x, algorithm='maxima') # needs sage.libs.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)
>>> from sage.all import * >>> integral(Integer(1)/(x**Integer(7)-Integer(1)), x, algorithm='maxima') # needs sage.libs.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)
integral(1/(x**7-1), x, algorithm='maxima') # needs sage.libs.maxima
We now show that floats are not converted to rationals automatically since we by default have
keepfloat: truein maxima.sage: integral(e^(-x^2), (x, 0, 0.1)) # needs sage.libs.maxima 0.05623145800914245*sqrt(pi)
>>> from sage.all import * >>> integral(e**(-x**Integer(2)), (x, Integer(0), RealNumber('0.1'))) # needs sage.libs.maxima 0.05623145800914245*sqrt(pi)
integral(e^(-x^2), (x, 0, 0.1)) # needs sage.libs.maxima
An example of an integral that fricas can integrate:
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)
>>> 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)
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: integrate(f(x), x) # long time # needs sage.libs.maxima 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))
>>> from sage.all import * >>> integrate(f(x), x) # long time # needs sage.libs.maxima 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))
integrate(f(x), x) # long time # needs sage.libs.maxima
The following definite integral is not found by maxima:
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') # long time # needs sage.libs.maxima integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4), x, 1, 2)
>>> 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') # long time # needs sage.libs.maxima integrate((x^4 - 3*x^2 + 6)/(x^6 - 5*x^4 + 5*x^2 + 4), x, 1, 2)
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') # long time # needs sage.libs.maxima
but is nevertheless computed:
sage: integrate(f(x), x, 1, 2) # long time # needs sage.libs.maxima -1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)
>>> from sage.all import * >>> integrate(f(x), x, Integer(1), Integer(2)) # long time # needs sage.libs.maxima -1/2*pi + arctan(8) + arctan(5) + arctan(2) + arctan(1/2)
integrate(f(x), x, 1, 2) # long time # needs sage.libs.maxima
Both fricas and sympy give the correct result:
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)
>>> 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)
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. If Giac is installed, this will be attempted automatically in the event that Maxima is unable to integrate the expression:
sage: # needs sage.libs.giac sage: result = integrate(abs(cos(x)), x, 0, 2*pi, algorithm='giac') ... sage: result 4 sage: # needs sage.libs.maxima sage: result = integrate(abs(cos(x)), x, 0, 2*pi) ... sage: result 4
>>> from sage.all import * >>> # needs sage.libs.giac >>> result = integrate(abs(cos(x)), x, Integer(0), Integer(2)*pi, algorithm='giac') ... >>> result 4 >>> # needs sage.libs.maxima >>> result = integrate(abs(cos(x)), x, Integer(0), Integer(2)*pi) ... >>> result 4
# needs sage.libs.giac result = integrate(abs(cos(x)), x, 0, 2*pi, algorithm='giac') result # needs sage.libs.maxima 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: a,b = var('a,b') sage: integrate(1/(x^3 *(a+b*x)^(1/3)), x) # needs sage.libs.maxima 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?
>>> from sage.all import * >>> a,b = var('a,b') >>> integrate(Integer(1)/(x**Integer(3) *(a+b*x)**(Integer(1)/Integer(3))), x) # needs sage.libs.maxima 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?
a,b = var('a,b') integrate(1/(x^3 *(a+b*x)^(1/3)), x) # needs sage.libs.maximaSo we just assume that \(a>0\) and the integral works:
sage: assume(a>0) sage: integrate(1/(x^3 * (a+b*x)^(1/3)), x) # needs sage.libs.maxima 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)
>>> from sage.all import * >>> assume(a>Integer(0)) >>> integrate(Integer(1)/(x**Integer(3) * (a+b*x)**(Integer(1)/Integer(3))), x) # needs sage.libs.maxima 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)
assume(a>0) integrate(1/(x^3 * (a+b*x)^(1/3)), x) # needs sage.libs.maxima