$\Large \int \frac{e^{log}\bigg(1+\frac{1}{x^2}\bigg)dx}{x^2+\frac{1}{x^2}}=$ $$(a)\;\frac{1}{\sqrt 2} \tan^{-1}\bigg(x+\frac{1}{x}\bigg)\qquad (b)\;\frac{1}{\sqrt 2}\tan^{-1}\bigg(\frac{x-\frac{1}{x}}{\sqrt 2}\bigg)\qquad(c)\;\sqrt 2 \tan^{-1}\bigg(x+\frac{1}{x}\bigg)\qquad(d)\;\frac{1}{\sqrt 2}\tan^{-1}\bigg(\frac{1}{x}-x\bigg)$$

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