# Evaluate the trigonometry integral $\int_0^{\frac{\pi}{4}}\frac{1+\cos x-x^{2}}{(1+x\sin x)\sqrt {1-x^{2}}}\,dx$

Find :

$$\int_0^{\frac{\pi}{4}}\frac{1+\cos x-x^{2}}{(1+x\sin x)\sqrt {1-x^{2}}}\,dx$$

I don't know how I starte & evaluate this integral

Wolfram alpha give $$=1,28553$$

My problem whene I use $$t=\cos x$$ I get $$\arccos x$$

Same problem with $$t=\sin x$$

So, either numerical integration or series expansions built at $$x=0$$. The latest would give $$\frac{1+\cos x-x^{2}}{(1+x\sin x)\sqrt {1-x^{2}}}=2-\frac{5 x^2}{2}+\frac{23 x^4}{8}-\frac{2323 x^6}{720}+\frac{17113 x^8}{4480}-\frac{5186177 x^{10}}{1209600}+O\left(x^{12}\right)$$ which would lead to something which evaluates as $$1.27486$$ (which is not very good).
Pushing the expansion up to $$O\left(x^{24}\right)$$ would give $$1.28486$$ (a lot of effort for a difference of 0.01 only !).