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Evaluate $$f(x) = \int_0^{\pi/2}\frac{1}{(1+x^2)(1+\tan{x})}dx$$

My attempt: I could not apply any standard method known to me to solve this integration. The only way I thought of is expressing $\tan(x)$ as an infinite series and expanding into a polynomial. But this will introduce approximation errors. $$f(x) = \int_0^{\pi/2}\frac{1}{(1+x^2)(x + \frac{x^3}{3}+\frac{2x^5}{15}+...)}dx$$ $$or, f(x) = \int_0^{\pi/2}{(1+x^2)^{-1}(x + \frac{x^3}{3}+\frac{2x^5}{15}+...)^{-1}}dx$$ Please let me know how to solve this problem.

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