I don't know any interesting bugs in symbolic algebra packages but I know a true, enlightening and entertaining story about something that looked like a bug but wasn't.$\def\sinc{\operatorname{sinc}}$
Define $\sinc x = (\sin x)/x$.
Someone found the following result in an algebra package:
$\int_0^\infty dx \sinc x = \pi/2$
They then found the following results:
$\int_0^\infty dx \sinc x \; \sinc (x/3)= \pi/2$
$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5)= \pi/2$
and so on up to
$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5) \; \cdots \; \sinc (x/13)= \pi/2$
So of course when they got:
$\int_0^\infty dx \sinc x \; \sinc (x/3) \sinc (x/5) \; \cdots \; \sinc (x/15)$$=
\frac{467807924713440738696537864469}{935615849440640907310521750000}\pi$
they knew they had to report the bug. The poor vendor struggled for a long time trying to fix it but eventually came to the stunning realisation that this result is correct.
These are now known as Borwein Integrals.
Best Answer
If you haven't already, you should try Sage.
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