What's wrong with this integral in mathematica?

• To: mathgroup at smc.vnet.net
• Subject: [mg60673] What's wrong with this integral in mathematica?
• From: "kiki" <lunaliu3 at yahoo.com>
• Date: Fri, 23 Sep 2005 04:20:49 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Hi all,

I am evaluating this integral:

\!\(Integrate[\(\[ExponentialE]\^\(\[ImaginaryI]\ a\ z\)\ Sinh[z]\)\/\((b +
\
Cosh[z])\)\^2, {z, \(-?\), ?}, Assumptions \[Rule] \ {a > 0, b > 1}]\)

After 1 hour running, it generates the following huge result: (after
simplification)

But even using my pencil and paper, do it manually, I should get much
simpler result...

Can anybody tell me what's wrong?

---------------------------------------------------

\!\(\((\((
1 + b - \@\(\(-1\) + b\^2\))\)\^\(\(-\[ImaginaryI]\)\ a\)\ \((1 + b +
\
\@\(\(-1\) + b\^2\))\)\^\(\(-\[ImaginaryI]\)\ a\)\ \((4\ a\ \((4 + 4\ \
\[ImaginaryI]\ a + a\^2 + \[ImaginaryI]\ a\^3)\)\ \@\(\(-1\) + b\^2\)\ \((1
+ \
b - \@\(\(-1\) + b\^2\))\)\^\(\[ImaginaryI]\ a\)\ \((1 +
b + \@\(\(-1\) + b\^2\))\)\^\(\[ImaginaryI]\ a\)\
AppellF1[1 - \[ImaginaryI]\ a, 2, 2, 2 - \[ImaginaryI]\ a,
\
\(-\(1\/\(b + \@\(\(-1\) + b\^2\)\)\)\), 1\/\(\(-b\) + \@\(\(-1\) +
b\^2\)\)] \
+ \((\[ImaginaryI] + a)\)\ \((4\ a\ \((\(-2\)\ \[ImaginaryI] + 3\ a + \
\[ImaginaryI]\ a\^2)\)\ b\ \@\(\(-1\) + b\^2\)\ \((1 + b - \@\(\(-1\) + \
b\^2\))\)\^\(\[ImaginaryI]\ a\)\ \((1 + b + \@\(\(-1\) + b\^2\))\)\^\(\
\[ImaginaryI]\ a\)\ AppellF1[2 - \[ImaginaryI]\ a, 2,
2, 3 - \[ImaginaryI]\ a, \(-\(1\/\(b + \@\(\(-1\) +
b\^2\)\)\)\),
1\/\(\(-
b\) + \@\(\(-1\) + b\^2\)\)] + \((2\ \[ImaginaryI] +
a)\)\ \((4\ \((2 + \[ImaginaryI]\ a)\)\ a\ \@\(\(-1\) + \
b\^2\)\ \((1 + b - \@\(\(-1\) + b\^2\))\)\^\(\[ImaginaryI]\ a\)\ \((1 + b +
\
\@\(\(-1\) +
b\^2\))\)\^\(\[ImaginaryI]\ a\)\
AppellF1[1 \
+ \[ImaginaryI]\ a, 2, 2, 2 + \[ImaginaryI]\ a, \(-\(1\/\(b + \@\(\(-1\) +
b\^2\)\)\)\), 1\/\(\(-b\) + \@\(\(-1\) +
b\^2\)\)] + \((\(-\[ImaginaryI]\) + a)\)\ \((4\ \[ImaginaryI]\ a\ b\
\@\(\(-1\
\) + b\^2\)\ \((1 + b - \@\(\(-1\) + b\^2\))\)\^\(\[ImaginaryI]\ a\)\ \((1 +
\
b + \@\(\(-1\) + b\^2\))\)\^\(\[ImaginaryI]\ a\)\ AppellF1[2 +
\[ImaginaryI]\
a, 2, 2, 3 + \[ImaginaryI]\ a, \(-\(1\/\(b +
\@\(\
\(-1\) + b\^2\)\)\)\), 1\/\(\(-b\) + \@\(\(-1\) + b\^2\)\)] - \[ImaginaryI]\
\
\((\(-2\)\ \[ImaginaryI] + a)\)\ \((\((1 + b - \@\(\(-1\) + b\^2\))\)\^\(\
\[ImaginaryI]\ a\)\ \((\(2 + 2\ b\)\/\(1 + b + \@\(\(-1\) + b\^2\)\))\)\^\(\
\[ImaginaryI]\ a\)\ \((1 + b + \@\(\(-1\) + b\^2\))\)\^\(\[ImaginaryI]\ a\)\
\
Hypergeometric2F1[\(-\[ImaginaryI]\)\ a, \(-\[ImaginaryI]\)\ a, 1 - \
\[ImaginaryI]\ a, \(\(-b\) + \@\(\(-1\) + b\^2\)\)\/\(\(-1\) - b +
\@\(\(-1\) \
+ b\^2\)\)] - \((1 + b - \@\(\(-1\) + b\^2\))\)\^\(\[ImaginaryI]\ a\)\ \
\((\(\(-2\) - 2\ b\)\/\(\(-1\) - b + \@\(\(-1\) + \
b\^2\)\))\)\^\(\[ImaginaryI]\ a\)\ \((1 + b + \@\(\(-1\) + b\^2\))\)\^\(\
\[ImaginaryI]\ a\)\ Hypergeometric2F1[\(-\[ImaginaryI]\)\ a,
\(-\[ImaginaryI]\
\)\ a, 1 - \[ImaginaryI]\ a, \(b + \@\(\(-1\) +
b\^2\)\)\/\(1 + b + \@\(\(-1\) + \
b\^2\)\)] + \((1 + b + \@\(\(-1\) + b\^2\))\)\^\(\[ImaginaryI]\ a\)\ \
Hypergeometric2F1[\[ImaginaryI]\ a, \[ImaginaryI]\ a, 1 + \[ImaginaryI]\ a,
\
\(b - \@\(\(-1\) + b\^2\)\)\/\(1 + b - \@\(\(-1\) + b\^2\)\)] - \((1 + b -
\@\
\(\(-1\) + b\^2\))\)\^\(\[ImaginaryI]\ a\)\ Hypergeometric2F1[\[ImaginaryI]\
\
a, \[ImaginaryI]\ a, 1 + \[ImaginaryI]\ a, \(b + \@\(\(-1\) + b\^2\)\)\/\(1
+
b + \@\(\(-1\) +
b\^2\)\)])\))\))\))\
\))\))\)/\((a\ \((4 + 5\ a\^2 + a\^4)\)\ \@\(\(-1\) + b\^2\))\)\)

```

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