Integration of a function involving algebraic and trigonometric functions

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.