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.

Define $\mathop{\mathrm{sinc}} x = (\sin x)/x$.

Someone found the following result in an algebra package:
$\int_0^\infty dx \mathop{\mathrm{sinc}} x/x = \pi/2$

They then found the following results:

$\int_0^\infty dx \mathop{\mathrm{sinc}} x \mathop{\mathrm{sinc}} (x/3)= \pi/2$

$\int_0^\infty dx \mathop{\mathrm{sinc}} x \mathop{\mathrm{sinc}} (x/3) \mathop{\mathrm{sinc}} (x/5)= \pi/2$

and so on up to

$\int_0^\infty dx \mathop{\mathrm{sinc}} x \mathop{\mathrm{sinc}} (x/3) \mathop{\mathrm{sinc}} (x/5) \ldots \mathop{\mathrm{sinc}} (x/13)= \pi/2$

So of course when they got:


$\int_0^\infty dx \mathop{\mathrm{sinc}} x \mathop{\mathrm{sinc}} (x/3) \mathop{\mathrm{sinc}} (x/5) \ldots \mathop{\mathrm{sinc}} (x/15)=$ $467807924713440738696537864469\pi/935615849440640907310521750000$

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](http://mathworld.wolfram.com/BorweinIntegrals.html).