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I got a solution I want to get inverse but it is not working in Mathematica or Rubi:

f[x_] := (a*ArcTanh[(a*x)/Sqrt[-d + b*x^2]])/(a^2 - b)(Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[-d + b*x^2]])/(a^2 - b) + (a*Log[d + (a^2 - b)*x^2])/(2*(a^2 - b)); 

InverseFunction[(a*ArcTanh[(a*x)/Sqrt[-d + b*x^2]])/(a^2 - b) -(Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[-d + b*x^2]])/(a^2 - b) + (a*Log[d + (a^2 - b)*x^2])/(2*(a^2 - b)), x]

Reduce[y == f[x], x]

All parameters are positive.

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  • $\begingroup$ Probably you consider a real function? If yes, FunctionDomain[f[x], x, Reals] shows several necessary conditions which must be fullfilled. Without further restriction Mathematica cannot evaluate the ` InverseFunction` $\endgroup$ Nov 10, 2021 at 21:29
  • $\begingroup$ Duplicates: (257680), (257851) -- You make the same syntax error in InverseFunction above as was corrected for you in (257680). $\endgroup$
    – Michael E2
    Nov 11, 2021 at 18:40
  • $\begingroup$ What makes you think this equation/inverse can be solved symbolically? $\endgroup$
    – Michael E2
    Nov 11, 2021 at 18:41

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