wien bridge oscillator transfer function
To determine the transfer function of this Wien-bridge oscillator, you can try the fast analytical circuits techniques or FACTs. This is the documented problem number 9 actually.

The principle is quite simple: you consider the transfer function denominator as a combination of the circuit time constants determined when the stimulus is turned off. Basically, the exercise consists of temporarily disconnected a capacitor (or an inductor) and "look" through its connecting terminals to determine the resistance driving the capacitor. For a circuit like this one, you can determine the denominator and the numerator without writing a single line of algebra. The basic circuit to look at is this one:



The stimulus in your circuit is the op-amp output while the response is the voltage across the grounded capacitor. You will then calculate the necessary gain the non-inverting gain will have to exhibit to exactly compensate the attenuation of the filter at the oscillation frequency. 

If you do the maths ok, then you should end-up with a low-entropy transfer function arranged in the following way: 

\$H(s)=H_{res}\frac{1}{1+Q(\frac{\omega_0}{s}+\frac{s}{\omega_0})}\$ 

which is the transfer function of a band-pass filter. In this expression, \$H_{res}\$ is the attenuation to be compensated by \$R_f\$ and \$R_i\$ to ensure sustained oscillations.

The complete Mathcad sheet is given below and shows how the expression perfectly matches the one obtained from brute-force algebra:



You can look at an introductory seminar taught at APEC 2016 which smoothly shows how FACTs work. When you've tried them, there is no turning back : )