Integrate[(1/L)*((Cos[2*Pi*t/L]+1)/2)*(E^((a-(I*2*Pi*f))*t)),{t,(-L/2),(L/2)}]

Manipulate[(4 (π^2) Sinh[(1/2) L (a-2 I f π)])/(L (a-2 I f π) ((a^2) (L^2)-4 I a f (L^2) π-4 (-1+(f^2) (L^2)) (π^2))),{L,{1}}]

ComplexExpand[((4 (π^2) Sinh[(1/2) (a-2 I f π)])/((a-2 I f π) ((a^2)-4 I a f π-4 (-1+(f^2)) (π^2))))]

D[ArcTan[ (-((4 (a^3) (π^2) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((16 a (π^4) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((48 a (f^2) (π^4) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((24 (a^2) f (π^3) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 f (π^5) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((32 (f^3) (π^5) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))))/(((24 (a^2) f (π^3) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 f (π^5) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((32 (f^3) (π^5) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((4 (a^3) (π^2) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((16 a (π^4) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((48 a (f^2) (π^4) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))))],f]

Manipulate[(-((-((4 (a^3) (π^2) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((16 a (π^4) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((48 a (f^2) (π^4) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((24 (a^2) f (π^3) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 f (π^5) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((32 (f^3) (π^5) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))) (((24 (a^2) f (π^4) Cos[f π] Cosh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 f (π^6) Cos[f π] Cosh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((32 (f^3) (π^6) Cos[f π] Cosh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-(24 (a^2) f (π^3) (32 (a^2) f (π^2)-16 f (π^2) ((a^2)-4 (-1+(f^2)) (π^2))) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (((16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))^2))-(32 f (π^5) (32 (a^2) f (π^2)-16 f (π^2) ((a^2)-4 (-1+(f^2)) (π^2))) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (((16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))^2))+(32 (f^3) (π^5) (32 (a^2) f (π^2)-16 f (π^2) ((a^2)-4 (-1+(f^2)) (π^2))) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (((16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))^2))-((192 (a^2) (f^2) (π^5) Cosh[(a/2)] Sin[f π])/(((((a^2)+4 (f^2) (π^2)))^2) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((256 (f^2) (π^7) Cosh[(a/2)] Sin[f π])/(((((a^2)+4 (f^2) (π^2)))^2) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((256 (f^4) (π^7) Cosh[(a/2)] Sin[f π])/(((((a^2)+4 (f^2) (π^2)))^2) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((24 (a^2) (π^3) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 (π^5) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((96 (f^2) (π^5) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-(4 (a^3) (π^2) (32 (a^2) f (π^2)-16 f (π^2) ((a^2)-4 (-1+(f^2)) (π^2))) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (((16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))^2))-(16 a (π^4) (32 (a^2) f (π^2)-16 f (π^2) ((a^2)-4 (-1+(f^2)) (π^2))) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (((16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))^2))+(48 a (f^2) (π^4) (32 (a^2) f (π^2)-16 f (π^2) ((a^2)-4 (-1+(f^2)) (π^2))) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (((16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))^2))-((32 (a^3) f (π^4) Cos[f π] Sinh[(a/2)])/(((((a^2)+4 (f^2) (π^2)))^2) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((128 a f (π^6) Cos[f π] Sinh[(a/2)])/(((((a^2)+4 (f^2) (π^2)))^2) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((384 a (f^3) (π^6) Cos[f π] Sinh[(a/2)])/(((((a^2)+4 (f^2) (π^2)))^2) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((96 a f (π^4) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((4 (a^3) (π^3) Sin[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((16 a (π^5) Sin[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((48 a (f^2) (π^5) Sin[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))))/(((((24 (a^2) f (π^3) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 f (π^5) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((32 (f^3) (π^5) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((4 (a^3) (π^2) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((16 a (π^4) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((48 a (f^2) (π^4) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))))^2)+(-((4 (a^3) (π^3) Cos[f π] Cosh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((16 a (π^5) Cos[f π] Cosh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((48 a (f^2) (π^5) Cos[f π] Cosh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+(4 (a^3) (π^2) (32 (a^2) f (π^2)-16 f (π^2) ((a^2)-4 (-1+(f^2)) (π^2))) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (((16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))^2))+(16 a (π^4) (32 (a^2) f (π^2)-16 f (π^2) ((a^2)-4 (-1+(f^2)) (π^2))) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (((16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))^2))-(48 a (f^2) (π^4) (32 (a^2) f (π^2)-16 f (π^2) ((a^2)-4 (-1+(f^2)) (π^2))) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (((16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))^2))+((32 (a^3) f (π^4) Cosh[(a/2)] Sin[f π])/(((((a^2)+4 (f^2) (π^2)))^2) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((128 a f (π^6) Cosh[(a/2)] Sin[f π])/(((((a^2)+4 (f^2) (π^2)))^2) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((384 a (f^3) (π^6) Cosh[(a/2)] Sin[f π])/(((((a^2)+4 (f^2) (π^2)))^2) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((96 a f (π^4) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-(24 (a^2) f (π^3) (32 (a^2) f (π^2)-16 f (π^2) ((a^2)-4 (-1+(f^2)) (π^2))) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (((16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))^2))-(32 f (π^5) (32 (a^2) f (π^2)-16 f (π^2) ((a^2)-4 (-1+(f^2)) (π^2))) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (((16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))^2))+(32 (f^3) (π^5) (32 (a^2) f (π^2)-16 f (π^2) ((a^2)-4 (-1+(f^2)) (π^2))) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (((16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))^2))-((192 (a^2) (f^2) (π^5) Cos[f π] Sinh[(a/2)])/(((((a^2)+4 (f^2) (π^2)))^2) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((256 (f^2) (π^7) Cos[f π] Sinh[(a/2)])/(((((a^2)+4 (f^2) (π^2)))^2) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((256 (f^4) (π^7) Cos[f π] Sinh[(a/2)])/(((((a^2)+4 (f^2) (π^2)))^2) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((24 (a^2) (π^3) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 (π^5) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((96 (f^2) (π^5) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((24 (a^2) f (π^4) Sin[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((32 f (π^6) Sin[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 (f^3) (π^6) Sin[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))))/(((24 (a^2) f (π^3) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 f (π^5) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((32 (f^3) (π^5) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((4 (a^3) (π^2) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((16 a (π^4) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((48 a (f^2) (π^4) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))))/(1+(((-((4 (a^3) (π^2) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((16 a (π^4) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((48 a (f^2) (π^4) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((24 (a^2) f (π^3) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 f (π^5) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((32 (f^3) (π^5) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))))^2)/(((((24 (a^2) f (π^3) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 f (π^5) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((32 (f^3) (π^5) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((4 (a^3) (π^2) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((16 a (π^4) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((48 a (f^2) (π^4) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))))^2)),{f,{0}}]

FullSimplify[((-((4 a (π^3) Cosh[(a/2)])/((((a^2)+4 (π^2)))^2))-((16 (π^5) Cosh[(a/2)])/(a ((((a^2)+4 (π^2)))^2)))+((24 (π^3) Sinh[(a/2)])/((((a^2)+4 (π^2)))^2))+((32 (π^5) Sinh[(a/2)])/((a^2) ((((a^2)+4 (π^2)))^2))))/(((4 a (π^2) Sinh[(a/2)])/((((a^2)+4 (π^2)))^2))+((16 (π^4) Sinh[(a/2)])/(a ((((a^2)+4 (π^2)))^2)))))]

Plot[π ((2/a)+((4 a)/((a^2)+4 (π^2)))-Coth[(a/2)]),{a,-10,10}]

FullSimplify[Sqrt[((-((4 (a^3) (π^2) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((16 a (π^4) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((48 a (f^2) (π^4) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((24 (a^2) f (π^3) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 f (π^5) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((32 (f^3) (π^5) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))))^2)+((((24 (a^2) f (π^3) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((32 f (π^5) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((32 (f^3) (π^5) Cosh[(a/2)] Sin[f π])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((4 (a^3) (π^2) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))+((16 a (π^4) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2))))-((48 a (f^2) (π^4) Cos[f π] Sinh[(a/2)])/(((a^2)+4 (f^2) (π^2)) (16 (a^2) (f^2) (π^2)+((((a^2)-4 (-1+(f^2)) (π^2)))^2)))))^2)]]

Manipulate[LogPlot[2 (Sqrt[2]) (π^2) (Sqrt[((-Cos[2 f π]+Cosh[a])/((a^6)+4 (a^4) (2+3 (f^2)) (π^2)+16 (a^2) (1+3 (f^4)) (π^4)+64 (f^2) (((-1+(f^2)))^2) (π^6)))]),{f,-10,10}],{a,0,10}]