I am facing integrals of the form
\begin{equation} I = \int \frac{\sin^{2}\left(x\right)}{\sin^{2}\left(x\right) + C\sin\left(2x\right) + C^{2}x^{2}}\, \mathrm{d}x \end{equation}
for which I could not find analytical solutions on Wolfram Alpha and in the Gradshteyn-Ryzhik book. Without success I tried "standard" techniques like integration by parts and substitution techniques like Weierstraß. Is anyone familiar in solving these kind of integrals? If not, does anyone know how to analytically solve integrals of the form
\begin{equation} I = \int \frac{\sin^{2}\left(x\right)}{\sin^{2}\left(x\right) + C^{2}x^{2}}\, \mathrm{d}x \end{equation}
or
\begin{equation} I = \int \frac{\sin\left(x\right)}{\sin\left(x\right) + Cx}\, \mathrm{d}x \end{equation}
For the simplest one $$I = \int \frac{\sin\left(x\right)}{\sin\left(x\right) + Cx}\,dx$$ you can write $$I=\sum_{n=0}^\infty (-1)^n\, C^{-(n+1)} \int \left(\frac{\sin (x)}{x}\right)^{n+1} \,dx$$ and the integrals $$J_n=\int \left(\frac{\sin (x)}{x}\right)^{n+1} \,dx$$ express in terms of sines, cosines, powers of $x$ and sine integrals.
For example $$J_2=\frac{1}{8} \left(9\, \text{Si}(3 x)-3 \,\text{Si}(x)-\frac{4 \sin ^2(x) (\sin (x)+3 x \cos (x))}{x^2}\right)$$
The problem is that the convergence of the summation could be a bit chaotic.