Problems with sympy.integrate

[AlexHaifa]

I would like to move from Mathematica to Sympy. Unfortunately, I couldn't calculate several integrals on Sympy, which Mathematica calculates easily. For example, Mathematica analytically integrate the Green function (gf) over X and Y

norm=sqrt(X**2+Y**2); gf=1/ norm

giving simple expression

expr=Y*log(X+ norm) + X*log(Y+ norm).

I was lucky enough to prove this result on Sympy, using differentiation and simplifications:

import sympy

X,  Y = sympy.symbols("X, Y",  real=True)

from sympy import  log, sqrt, diff

norm=sqrt(X**2+Y**2)

expr=Y*log(X+ norm) + X*log(Y+ norm)

sympy.radsimp(sympy.simplify(diff(expr, X, Y)))-1/norm =0

My naïve attempt to calculate the integral

sympy.integrate(1/norm, X, Y)

failed, since Sympy return the following expression

X*Integral(Abs(Y)*asinh(Abs(X)/Abs(Y))/Y, Y)/Abs(X)

Then I tried to simplify calculation of the integral,   subdividing it to two following integrals

Int1= sympy.integrate(1/norm, X)

Int2= sympy.integrate(Int1, Y).

Sympy calculated Int1 as

X*Abs(Y)*asinh(Abs(X)/Abs(Y))/(Y*Abs(X))

 

And Int2 returned unevaluated in the form

X*Integral(Abs(Y)*asinh(Abs(X)/Abs(Y))/Y, Y)/Abs(X)

 

I recalculate Int1 by Mathematica and have got results expressed in terms of logarithmic function Int1=log(X+ norm)

(rather than asinh as Sympy does). And, finally, I have calculated

sympy.integrate(log(X+ norm), Y)

and have got results in the form

Y*log(X + sqrt(X**2 + Y**2)) + Y*Abs(X)*asinh(Abs(Y)/Abs(X))/Abs(Y) - Y

I feel that this result is equal to the Mathematica result Y*log(X+ norm) + X*log(Y+ norm), but failed to prove it.

May be somebody knows, how to calculate integrals  

sympy.integrate(1/norm, X), sympy.integrate(1/norm, X, Y)

in terms of logarithmic functions?

[Aaron Meurer]

You can convert asinh to log using rewrite(), like expr.rewrite(log).
It might take some further simplification to get it in the exact form
you want.

Aaron Meurer

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