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I have the following code:

ser[x_] = FourierSeries[(\[Pi]^2 + a)/(3 x^2 + a), x, 10] // N // Chop

It gives me some series, which I then try to plot. And surprisingly, the result isn't even similar to the function I passed to FourierSeries[]: for comparison, I've used this code:

Plot[{ser[x], 1000 (\[Pi]^2 + a)/(3 x^2 + a)}, {x, -\[Pi], \[Pi]}]

Fourier transform for function given in documentation works correctly, while for this one doesn't. I've tried using directly the formula given in documentation as default formula (used NIntegrate[]), and that gives me expected results.
What have I done wrong? Is this a bug?

Addition to answer comments:

As one can see, the Fourier coefficients computed via NIntegrate[] are quite different from ones generated by FourierCoefficient[].

Update:
As pointed in comments, setting GenerateConditions->True appears to yield correct result, not generating any conditions though. Why should this be needed?

Bug introduced in Version 8 or earlier, and persisting through 12.1.

I have the following code:

ser[x_] = FourierSeries[(π^2 + a)/(3 x^2 + a), x, 10] // N // Chop

It gives me some series, which I then try to plot. And surprisingly, the result isn't even similar to the function I passed to FourierSeries[]: for comparison, I've used this code:

Plot[{ser[x], 1000 (π^2 + a)/(3 x^2 + a)}, {x, -π, π}]

Fourier transform for function given in documentation works correctly, while for this one doesn't. I've tried using directly the formula given in documentation as default formula (used NIntegrate[]), and that gives me expected results.
What have I done wrong? Is this a bug?

Addition to answer comments:

As one can see, the Fourier coefficients computed via NIntegrate[] are quite different from ones generated by FourierCoefficient[].

Update:
As pointed in comments, setting GenerateConditions->True appears to yield correct result, not generating any conditions though. Why should this be needed?

I have the following code:

ser[x_] = FourierSeries[(\[Pi]^2 + a)/(3 x^2 + a), x, 10] // N // Chop

It gives me some series, which I then try to plot. And surprisingly, the result isn't even similar to the function I passed to FourierSeries[]: for comparison, I've used this code:

Plot[{ser[x], 1000 (\[Pi]^2 + a)/(3 x^2 + a)}, {x, -\[Pi], \[Pi]}]

Fourier transform for function given in documentation works correctly, while for this one doesn't. I've tried using directly the formula given in documentation as default formula (used NIntegrate[]), and that gives me expected results.
What have I done wrong? Is this a bug?

Addition to answer comments:

As one can see, the Fourier coefficients computed via NIntegrate[] are quite different from ones generated by FourierCoefficient[].

I have the following code:

ser[x_] = FourierSeries[(\[Pi]^2 + a)/(3 x^2 + a), x, 10] // N // Chop

It gives me some series, which I then try to plot. And surprisingly, the result isn't even similar to the function I passed to FourierSeries[]: for comparison, I've used this code:

Plot[{ser[x], 1000 (\[Pi]^2 + a)/(3 x^2 + a)}, {x, -\[Pi], \[Pi]}]

Fourier transform for function given in documentation works correctly, while for this one doesn't. I've tried using directly the formula given in documentation as default formula (used NIntegrate[]), and that gives me expected results.
What have I done wrong? Is this a bug?

Addition to answer comments:

As one can see, the Fourier coefficients computed via NIntegrate[] are quite different from ones generated by FourierCoefficient[].

Update:
As pointed in comments, setting GenerateConditions->True appears to yield correct result, not generating any conditions though. Why should this be needed?