<s>[INST] Definite integrals of fractions of trig functions $$
 \int _{\pi/4}^{\pi/3}\sin(x)\frac{\sin(x)+\cos^2(x)}{\sqrt{2}\sin(x)+\cos(x)}
\,\mathrm{d}x
$$
How do you solve for this integral?
I was considering using Weierstrass substitution or manipulating the integrand with respect to $\sec(x)$ or $\csc(x)$ but couldn't yield satisfactory results...
 [/INST] Using Weierstrass substitution, I suppose that you did arrive at
$$-4\int \frac{t \left(t^4+2 t^3-2 t^2+2 t+1\right) } {\left(t^2+1\right)^3 \left(t^2-2 \sqrt{2} t-1\right) }\,dt=-4\int \frac{t \left(t^4+2 t^3-2 t^2+2 t+1\right) } {\left(t^2+1\right)^3 \left(t+\sqrt 3-\sqrt 2\right)\left(t-\sqrt 3-\sqrt 2\right) }\,dt$$
Using partial fraction decomposition, the integrand is
$$\frac{-2 \sqrt{2} t-3 \sqrt{2}-4}{18 \sqrt{2} \left(t^2+1\right)}+\frac{\sqrt{2}
   t-2 t+\sqrt{2}+4}{3 \sqrt{2} \left(t^2+1\right)^2}-\frac{\sqrt{2} \left(\sqrt{2}
   t+2\right)}{3 \left(t^2+1\right)^3}+$$ $$\frac{3-2 \sqrt{3}}{36 \sqrt{3}
   \left(-t-\sqrt{3}+\sqrt{2}\right)}+\frac{-3-2 \sqrt{3}}{36 \sqrt{3}
   \left(-t+\sqrt{3}+\sqrt{2}\right)}$$ which seems to be more than practicable.
</s>