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You are right, when we add the results we get zero. But this don't make sense. The subtraction should be zero, since L>0. So, this is something that I don't formulate very well in Sage? Or is something about my adaptation to Sage?

I'm making the following calculations:

X,Y,L=var('X,Y,L')

assume(L>0)

F(L)=integrate(integrate(log((X)^2+(Y)^2),Y,L/2,3*L/2),X,-L/2,L/2)

Maxima requested tor assume(4*X^2+L^2-4>0) and for assume(L-2>0), so I run the following

X,Y,L=var('X,Y,L')

assume(L>0)

assume(4*X^2+L^2-4>0)

assume(L-2>0)

F(L)=integrate(integrate(log((X)^2+(Y)^2),Y,L/2,3*L/2),X,-L/2,L/2)

print(F)

and the result is

-1/4piL^2 + 1/2L^2(arctan(3) - 9arctan(1/3) - 2) + 9L^2arctan(1/3) + 3/2L^2log(5/2L^2) - 1/2L^2log(1/2L^2) - 2L^2

After this, I change the limits of the integration

X,Y,L=var('X,Y,L')

assume(L>0)

assume(4*X^2+L^2-4>0)

assume(L-2>0)

F(L)=integrate(integrate(log((X)^2+(Y)^2),Y,-3*L/2,-L/2),X,-L/2,L/2)

print(F)

and the result is

1/4piL^2 - 1/2L^2(arctan(3) - 9arctan(1/3) - 2) - 9L^2arctan(1/3) - 3/2L^2log(5/2L^2) + 1/2L^2log(1/2L^2) + 2L^2

If we subtract this to results and considering L=10 The result should be Zero, but the result is very different

Z(L)=(-1/4piL^2 + 1/2L^2(arctan(3) - 9arctan(1/3) - 2) + 9L^2arctan(1/3) + 3/2L^2log(5/2L^2) - 1/2L^2log(1/2L^2) - 2L^2)-(1/4piL^2 - 1/2L^2(arctan(3) - 9arctan(1/3) - 2) - 9L^2arctan(1/3) - 3/2L^2log(5/2L^2) + 1/2L^2log(1/2L^2) + 2L^2)

Z(10).n()

922.636418333173

I know that the result should be zero from the math and I also use the software Mathematica.

The question is: I'm making something wrong? Or this is a well known problem of Sage?

Thanks, I really didn't realize about that difference between "log" and "Log" since mathematica keep producing results.

The minus sign in Sage is really very weird.

Thanks.

In Phd thesis, I'm having some trouble to calculating some tricky integrals, Sage and Mathematica show different results. To understand what goes I have calculated a simple integral.

At Sage:

var('y')
assume(y>0)
integral(log(sqrt(x^2+y^2),10),x,-10,-1)

Result: -1/2(2yarctan(10/y) - 2yarctan(1/y) + 10log(y^2 + 100) - log(y^2 +1) - 18)/log(10)

At Mathematica:

Assuming[y > 0, Integrate[log10 (Sqrt[x ^2 + y^2]), {x, -10, -1}]]

Result: 1/2 log10 (-Sqrt[1 + y^2] + 10 Sqrt[100 + y^2] + y^2 Log[(10 + Sqrt[100 + y^2])/(1 + Sqrt[1 + y^2])])

If we plot the results the output in the interval (y,-8,8), the plots will be very different. I'm making any mistake? Why this happens? Which one should I believe?