<s>[INST] Evaluating $\int \frac{x \sqrt{2\sin(x^2+1) - \sin2(x^2+1)}}{2 \sin(x^2+1)+\sin2(x^2+1)} dx$ The question is to evaluate $$\int \frac{x \sqrt{2\sin(x^2+1) - \sin2(x^2+1)}}{2 \sin(x^2+1)+\sin2(x^2+1)} dx$$ if $x^2 \neq (n \pi-1) \forall n\in N$
I tried to rewrite the integral as $$\int \frac{x\sqrt{(2\sin(x^2+1))(1-\cos(x^2+1))}}{(2\sin(x^2+1))(1+\cos(x^2+1))}$$
If there would have been a radical sign in denominator too the problem would be quite simple but I couldn't understand how to proceed from here.Any ideas?
 [/INST] Hint:
Let $x^2+1=u$ then your integral simplifies to
$$\int\dfrac{\sqrt{8}}{8}\dfrac{du}{\left(\cos\frac{u}{2}\right)^\frac52}$$
now let $\sin\dfrac{u}{2}=w$.
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