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Question: Why can't the Maple simplifier combine the dilogarithm function?

Question:Why can't the Maple simplifier combine the dilogarithm function?

sursumCorda 1099 Maple 2023
simplify dilog
April 14 2023
0

The result of expr__1 is said to be the Catalan constant, but unfortunately, Maple® only returns a lengthy output, so I have to apply the simplify command to get a shorter (and equivalent) form of it. However, I find that these do not work here: 

> 

restart;
> 

expr__1 := expand(value(student[Doubleint](sec(x+y)*sec(x-y)/(sec(x)*sec(y)), x = 0 .. (1/4)*Pi, y = 0 .. (1/4)*Pi)))
> 

MmaTranslator:-Mma:-Chop(evalf(expr__1-Catalan))

((1/2)*I)*dilog(1-(1/2)*2^(1/2)+((1/2)*I)*2^(1/2))+((1/2)*I)*dilog(1-(1/2)*2^(1/2)-((1/2)*I)*2^(1/2))-((1/2)*I)*dilog(1+(1/2)*2^(1/2)+((1/2)*I)*2^(1/2))-((1/2)*I)*dilog(1+(1/2)*2^(1/2)-((1/2)*I)*2^(1/2))-((1/8)*I)*Pi^2+Catalan

  

0

 (1)
> 

simplify(expr__1, size = false)-Catalan; Physics:-Simplify(expr__1)-Catalan; simplify(expr__1-Catalan, size = false); Physics:-Simplify(expr__1-Catalan); verify(expr__1, Catalan, equal); is(expr__1 = Catalan); verify(expr__1-Catalan, 0, equal); is(expr__1-Catalan, 0)

((1/2)*I)*dilog(1-(1/2)*2^(1/2)+((1/2)*I)*2^(1/2))+((1/2)*I)*dilog(1-(1/2)*2^(1/2)-((1/2)*I)*2^(1/2))-((1/2)*I)*dilog(1+(1/2)*2^(1/2)+((1/2)*I)*2^(1/2))-((1/2)*I)*dilog(1+(1/2)*2^(1/2)-((1/2)*I)*2^(1/2))-((1/8)*I)*Pi^2

  

((1/2)*I)*dilog(1-(1/2)*2^(1/2)+((1/2)*I)*2^(1/2))+((1/2)*I)*dilog(1-(1/2)*2^(1/2)-((1/2)*I)*2^(1/2))-((1/2)*I)*dilog(1+(1/2)*2^(1/2)+((1/2)*I)*2^(1/2))-((1/2)*I)*dilog(1+(1/2)*2^(1/2)-((1/2)*I)*2^(1/2))-((1/8)*I)*Pi^2

  

((1/8)*I)*(-Pi^2+4*dilog(1-(1/2)*2^(1/2)+((1/2)*I)*2^(1/2))+4*dilog(1-(1/2)*2^(1/2)-((1/2)*I)*2^(1/2))-4*dilog(1+(1/2)*2^(1/2)+((1/2)*I)*2^(1/2))-4*dilog(1+(1/2)*2^(1/2)-((1/2)*I)*2^(1/2)))

  

((1/8)*I)*(-Pi^2+4*dilog(1-(1/2)*2^(1/2)+((1/2)*I)*2^(1/2))+4*dilog(1-(1/2)*2^(1/2)-((1/2)*I)*2^(1/2))-4*dilog(1+(1/2)*2^(1/2)+((1/2)*I)*2^(1/2))-4*dilog(1+(1/2)*2^(1/2)-((1/2)*I)*2^(1/2)))

  

FAIL

  

FAIL

  

FAIL

  

FAIL

 (2)
> 

expr__2 := expand(convert(expr__1, polylog, simplifier = NONE))

((1/2)*I)*polylog(2, (1/2)*2^(1/2)-((1/2)*I)*2^(1/2))+((1/2)*I)*polylog(2, (1/2)*2^(1/2)+((1/2)*I)*2^(1/2))-((1/2)*I)*polylog(2, -((1/2)*I)*2^(1/2)-(1/2)*2^(1/2))-((1/2)*I)*polylog(2, -(1/2)*2^(1/2)+((1/2)*I)*2^(1/2))-((1/8)*I)*Pi^2+Catalan

 (3)
> 

simplify(expr__2, size = false)-Catalan; Physics:-Simplify(expr__2)-Catalan; simplify(expr__2-Catalan, size = false); Physics:-Simplify(expr__2-Catalan); verify(expr__2, Catalan, equal); is(expr__2 = Catalan); verify(expr__2-Catalan, 0, equal); is(expr__2-Catalan, 0)

0

  

((1/2)*I)*polylog(2, (1/2-(1/2)*I)*2^(1/2))+((1/2)*I)*polylog(2, (1/2+(1/2)*I)*2^(1/2))-((1/2)*I)*polylog(2, (-1/2-(1/2)*I)*2^(1/2))-((1/2)*I)*polylog(2, (-1/2+(1/2)*I)*2^(1/2))-((1/8)*I)*Pi^2

  

0

  

-((1/8)*I)*(Pi^2+4*polylog(2, (-1/2+(1/2)*I)*2^(1/2))+4*polylog(2, (-1/2-(1/2)*I)*2^(1/2))-4*polylog(2, (1/2+(1/2)*I)*2^(1/2))-4*polylog(2, (1/2-(1/2)*I)*2^(1/2)))

  

true

  

true

  

true

  

true

 (4)
> 

NULL

Download Unable_to_simplify_expressions_containing_dilog.mw

(By the way, Mathematica's Integrate cannot compute this double integral explicitly.)
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