Here is the short version:
restart;
integrand:=1/(exp(x^3/3)*x^2);
y2:= int( integrand, x);
Maple gives
1/9*3^(2/3)*(-9/10*x^2*3^(2/3)/(x^3)^(1/3)*exp(-1/6*x^3)* WhittakerM(1/3,5/6,1/3*x^3)-9/2/x^4*3^(2/3)*(x^3+2)/(x^3)^(1/3)* exp(-1/6*x^3)*WhittakerM(4/3,5/6,1/3*x^3))
but this does not differentiate back to the integrand. I tried
simplify and assumptions. No luck.
I think the Maple antiderivative is wrong but I am still not sure.
On Saturday, January 15, 2022 at 12:19:28 PM UTC-5, Nasser M. Abbasi wrote: ...
Here is the short version:
restart;
integrand:=1/(exp(x^3/3)*x^2);
y2:= int( integrand, x);
Maple gives
1/9*3^(2/3)*(-9/10*x^2*3^(2/3)/(x^3)^(1/3)*exp(-1/6*x^3)* WhittakerM(1/3,5/6,1/3*x^3)-9/2/x^4*3^(2/3)*(x^3+2)/(x^3)^(1/3)* exp(-1/6*x^3)*WhittakerM(4/3,5/6,1/3*x^3))
but this does not differentiate back to the integrand. I tried
simplify and assumptions. No luck.
I think the Maple antiderivative is wrong but I am still not sure.Using Maple 2021.1,
restart;
integrand:=1/(exp(x^3/3)*x^2):
y2:= int( integrand, x):
1/9*3^(2/3)*(-9/10*x^2*3^(2/3)/(x^3)^(1/3)*exp(-1/6*x^3)*WhittakerM(1/3,5/6,1/3
*x^3)-9/2/x^4*3^(2/3)*(x^3+2)/(x^3)^(1/3)*exp(-1/6*x^3)*WhittakerM(4/3,5/6,1/3*
x^3))
expand(simplify(convert(diff(y2,x),compose,
hypergeom,StandardFunctions))):
exp(-1/3*x^3)/x^2
As I was verifying solution to an ode. Even thought everything I did
was correct, Maple was saying my solution is not correct (using odetest).
I zoomed into the problem and it is comming from result of integrate step
I did using Maple.
Here is the short version:
------------
restart;
integrand:=1/(exp(x^3/3)*x^2);
y2:= int( integrand, x);
-----------
Maple gives
1/9*3^(2/3)*(-9/10*x^2*3^(2/3)/(x^3)^(1/3)*exp(-1/6*x^3)* WhittakerM(1/3,5/6,1/3*x^3)-9/2/x^4*3^(2/3)*(x^3+2)/(x^3)^(1/3)* exp(-1/6*x^3)*WhittakerM(4/3,5/6,1/3*x^3))
but this does not differentiate back to the integrand. I tried
simplify and assumptions. No luck.
I think the Maple antiderivative is wrong but I am still not sure.
I tried the same thing in Mathematica,
and it gives result using incomplete Gamma function, not WhittakerM,
and it does differentiate back to the integrand:
-------------
ClearAll[x]
integrand = 1/(Exp[x^3/3]*x^2)
y2 = Integrate[integrand, x]
---------
-(((x^3)^(1/3) Gamma[-(1/3),x^3/3])/(3 3^(1/3) x))
And now
D[y2, x] // Simplify
gives
E^(-(x^3/3))/x^2
Which is the integrand. So its results is correct.
I do not know the relation between incomplete Gamma and WhittakerM
to try to translate Maple's result to Gamma and compare better.
question is: Is Maple's int() result correct?
Mathematica is V 13. Maple is 2021.2. All on windows 10
--Nasser
On Sunday, 16 January 2022 at 04:19:28 UTC+11, Nasser M. Abbasi wrote:
As I was verifying solution to an ode. Even thought everything I did
was correct, Maple was saying my solution is not correct (using odetest).
I zoomed into the problem and it is comming from result of integrate step
I did using Maple.
Here is the short version:
------------
restart;
integrand:=1/(exp(x^3/3)*x^2);
y2:= int( integrand, x);
-----------
Maple gives
1/9*3^(2/3)*(-9/10*x^2*3^(2/3)/(x^3)^(1/3)*exp(-1/6*x^3)* WhittakerM(1/3,5/6,1/3*x^3)-9/2/x^4*3^(2/3)*(x^3+2)/(x^3)^(1/3)* exp(-1/6*x^3)*WhittakerM(4/3,5/6,1/3*x^3))
but this does not differentiate back to the integrand. I tried
simplify and assumptions. No luck.
I think the Maple antiderivative is wrong but I am still not sure.
I tried the same thing in Mathematica,
and it gives result using incomplete Gamma function, not WhittakerM,
and it does differentiate back to the integrand:
-------------
ClearAll[x]
integrand = 1/(Exp[x^3/3]*x^2)
y2 = Integrate[integrand, x]
---------
-(((x^3)^(1/3) Gamma[-(1/3),x^3/3])/(3 3^(1/3) x))
And now
D[y2, x] // Simplify
gives
E^(-(x^3/3))/x^2
Which is the integrand. So its results is correct.
I do not know the relation between incomplete Gamma and WhittakerM
to try to translate Maple's result to Gamma and compare better.
question is: Is Maple's int() result correct?
Mathematica is V 13. Maple is 2021.2. All on windows 10
--Nasserthese are too complicated. input your formula into mathHand.com, click the integrate button.
or click the link: http://server.drhuang.com/input/?guess=integrate%281%2F%28exp%28x%5E3%2F3%29*x%5E2%29%29&inp=1%2F%28exp%28x%5E3%2F3%29*x%5E2%29
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