How to do FourierSeries of this function?

I try to do FourierSeries in Mathematica but somehow this code runs forever...

Could someone help? Thank you.

FourierSeries[(a^2 b Sin[x])/( b^2 (Cos[x])^2 + a^2 (Sin[x])^2 )^(3/2),x,2]

First, this is an odd function, so FourierSeries is FourierSinSeries. Second, up to constant multipliers , the following does the job.

FourierSinSeries[(Sin[x])/(1 + c^2 (Sin[x])^2)^(3/2), x, 2, Assumptions -> c > 0]

(1/\[Pi])2 ((-EllipticE[-c^2] + (1 + c^2) EllipticK[-c^2])/( c^2 + c^4) + (-EllipticE[c^2/(1 + c^2)] + EllipticK[c^2/(1 + c^2)])/(c^2 Sqrt[1 + c^2])) Sin[x]

and/or

FourierSinSeries[(Sin[x])/(1 + c^2 (Cos[x])^2)^(3/2), x, 2, Assumptions -> c > 0]

(1/(c^2 \[Pi]))2 (EllipticE[-c^2] - EllipticK[-c^2] + ((1 + c^2) EllipticE[c^2/(1 + c^2)] - EllipticK[c^2/(1 + c^2)])/Sqrt[1 + c^2]) Sin[x]

• Just to compare. The code of Maple with(OrthogonalExpansions): assuming([FourierSeries(sin(x)/(1+c^2*sin(x)^2)^(3/2), x = -Pi .. Pi, 2)], [c > 0]); performs $${\frac {\sin \left( x \right) }{\pi} \left( 4\,{\frac {1}{ \left( {c}^ {2}+1 \right) ^{3/2}}{\it EllipticK} \left( {\frac {c}{\sqrt {{c}^{2}+ 1}}} \right) }+4\,{\frac {1}{ \left( {c}^{2}+1 \right) ^{3/2}{c}^{2}}{ \it EllipticK} \left( {\frac {c}{\sqrt {{c}^{2}+1}}} \right) }-4\,{ \frac {1}{\sqrt {{c}^{2}+1}{c}^{2}}{\it EllipticE} \left( {\frac {c}{ \sqrt {{c}^{2}+1}}} \right) } \right) } .$$ ` Commented Nov 4, 2022 at 18:55