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.