"Oldest" bug in computer algebra system? The goal of this question is to find an error in a computation by a computer algebra system where the 'correct' answer (complete with correct reasoning to justify the answer) can be found in the literature.  Note that the system must claim to be able to perform that computation, not implementing a piece of (really old) mathematics is sad, but is a different topic.
From my knowledge of the field, there are plenty of examples of 19th century mathematics where today's computer algebra system get the wrong answer.  But how far back can we go?

Let me illustrate what I mean.  James Bernoulli in letters to Leibniz (circa 1697-1704) wrote that [in today's notation, where I will assume that $y$ is a function of $x$ throughout] he could not find a closed-form to $y' = y^2 + x^2$.  In a letter of Nov. 15th, 1702, he wrote to Leibniz that he was however able to reduce this to a 2nd order LODE, namely $y''/y = -x^2$.  Maple can find (correct) closed-forms for both of these differential equations, in terms of Bessel functions.
An example that is 'sad' but less interesting is
$$r^{n+1}\int_0^{\pi}\cos(r\rho \cos (\omega))\sin(\omega)^{2n+1}d\omega$$
with $n$ assumed to be a positive integer, $r>0$ and $\rho$ real; this can be evaluated as a Bessel functions but, for example, Maple can't.  Poisson published this result in a long memoir of 1823.
One could complain that (following Schloemilch, 1857) that he well knew that
$$J_n(z) = \sum_{0}^{\infty} \frac{(-1)^m(z/2)^{n+2*m}}{m!(n+m)!}$$
Maple seems to think that this sum is instead $J_n(z)\frac{\Gamma(n+1)}{n!}$, which no mathematician would ever write down in this manner.
Another example which gets closer to a real bug is that  Lommel in 1871 showed that the Wronskian of $J_{\nu}$ and $J_{-\nu}$ was $-2\frac{sin(\nu\pi)}{\nu z}$.  Maple can compute the Wronskian, but it cannot simplify the result to $0$.  This can be transformed into a bug by using the resulting expression in a context where we force the CAS to divide by it.
For a real bug, consider
$$\int_{0}^{\infty} t^{-\lambda} J_{\mu}(at) J_{\nu}(bt)$$
as investigated by Weber in 1873.  Maple returns an unconditional answer, which a priori looks fine.  If, however, the same question is asked but with $a=b$, no answer is returned!  What is going on?  Well, in reality that answer is only valid for one of $0\lt a\lt b$ or $0\lt b \lt a$.  But it turns out (as Watson explains lucidly on pages 398-404 of his master treatise on Bessel functions, this integral is discontinuous for $a=b$.  Actually, the answer given is also problematic for $\lambda=\mu=0, \nu=1$.  And for the curious, the answer given is
$$\frac{2^{-\lambda}{a}^{\lambda-1-\nu}{b}^{\nu}
\Gamma  \left( 1/2\nu+1/2\mu-1/2\lambda+1/2 \right)} { 
\Gamma\left( 1/2\mu+1/2\lambda+1/2-1/2\nu\right) \Gamma  \left( \nu+1 \right)}
{F(1/2-1/2\mu-1/2\lambda+1/2\nu,1/2\nu+1/2\mu-1/2\lambda+1/2;\nu+1;{\frac {{b}^{2}}{{a}^{2}}})}
$$

EDIT: I first asked this question when the MO community was much smaller.  Now that it has grown a lot, I think it needs a second go-around.  A lot of mathematicians use CASes routinely in their work, so wouldn't they be interested to know the 'age' gap between human mathematics and (trustable) CAS mathematics?
 A: I'm not sure I completely understand the question, but many versions of Maple give the wrong answer when counting the number of partitions of $n$, for some $n$.  Obviously, mathematicians have known how to do this since at least Euler. (One could argue that mathematicians have known how to count for a very long time, indeed.)
A: According to Wolfram Alpha and the tables in [2], $\pi(10^{10}) = 455, 052, 511$. Nevertheless, in Don Zagier's paper listed below we find that $\pi(10^{10}) = 455, 052, 512$.
Wonder whether someone has already noted this discrepancy between the sources elsewhere. Naturally, the discrepancy implies the existence of a bug in either the routines of Zagier or in WA's implementation of the prime counting function. I don't think that it's only a typo in Zagier' note because, if my memory serves me right, there are some other texts in the literature that endorse the computations of Zagier (for instance, see [1, page 7].).
References
[1] A. E. Ingham. The distribution of prime numbers. Cambridge Mathematical Library, 1934 (Reissued in 1990).
[2] H. Riesel. Prime Numbers and Computer Methods for Factorization. Birkhäuser, Second Edition, 1994. 
[3] D. Zagier. The first 50 million primes. Math. Intelligencer, 0 (1977).
A: In mathematica, if you look at the dirichlet characters modulo 4, you don't actually get the characters.
A: In Minkowski space-time one expects $$\epsilon_{ijkl}\epsilon_{i^\prime j^\prime k^\prime l^\prime}g^{ii^\prime}g^{jj^\prime}g^{kk^\prime}g^{ll^\prime}=-24, \tag{1}$$
where $\epsilon_{ijkl}$ is the Levi-Civita tensor and $g^{ij}$ represents the metric. However, if you (uncritically) calculate the l.h.s of (1) in the symbolic manipulation system FORM using FixIndex statement to assign specific values to selected diagonal elements of the Kronecker delta, which by default represents the metric, you get +24, not -24. 
In fact, this is not a bug but a subtlety of inner workings of FORM and the users were warned that one can try to change the behaviour of the Kronecker delta a bit but "this is dangerous and needs, in addition to a good
understanding of what is happening, good testing to make sure that what the
user wants is indeed what does happen":
http://www.nikhef.nl/~form/maindir/documentation/reference/online/
Interestingly, this subtlety of manipulations with the Levi-Civita tensor in FORM led to a long lasted sign error in the calculations of the pion-pole dominant term in the hadronic light-by-light scattering contribution to the muon anomalous magnetic moment, and the source of this error was discovered only when some later time calculations of the same quantity, based on the REDUCE Computer Algebra System, gave an oposite sign: http://arxiv.org/abs/hep-ph/0112102 (Comment on the sign of the pseudoscalar pole contribution to the
muon g-2, by Masashi Hayakawa and Toichiro Kinoshita).
I think the following paragraph in the above cited FORM manual gives a very good advice how modern computer algebra systems should be used:
"As in the Zen saying: 
To the beginning student mountains are mountains and water is water. To the advanced student mountains stop being mountains and water stops being water. 
To the master mountains are mountains again and water is water again. 
Of course the modern master also checks that what he expects the system to do, is indeed what the system does."
A: I don't know if you mean this but have a look here (there some bugs that seem to be quite elementary):
- http://www.walkingrandomly.com/?p=801
- http://www.walkingrandomly.com/?p=578
- http://www.walkingrandomly.com/?p=88
- ...search for "bug" on this site
Hope this helps
A: If I recall correctly from ~30 years ago, on the Apple ][ the calculation 7^2 would return 49.0001. More than 100 years ago (or even 100 years before that), mathematicians already knew that the square of an integer is an integer.
