Completely stuck at integrating $\int x\sqrt{\cos x}\,dx$ I am currently learning Calculus and I just started studying Integrals. I want to find the general indefinite integral $$\int x\sqrt{\cos x}\,dx$$
I have tried using some substitution such as letting $u=\cos x$ or $u=\sqrt x$, etc. but it seems like none of them is effective here.
I have also tried using online integral calculators to solve this question, but all I got is something like "steps are currently not supported for this problem."
Since even an integral calculator couldn't handle this, I wondered if this question can be solved. Therefore I came back to MathSE to seek help.
Is it solvable? If so, then how? Thanks in advance.
 A: Another solution kindly suggested by @Raymond Manzoni
$$I=\int x\sqrt{\cos (x)}\,dx=\int x\sqrt{1-2 \sin ^2\left(\frac{x}{2}\right)}\,dx$$
Let $x=2y$
$$I=4 \int y \sqrt{1-2 \sin ^2(y)}\,dy$$ Integration by parts
$$u=y \qquad v'=\sqrt{1-2 \sin ^2(y)}$$ gives
$$\frac 14 I=y\,E(y|2)-\int E(y|2)\,dy$$
Using the expansion
$$E(y|2)=y-\frac{y^3}{3}-\frac{y^5}{30}-\frac{19 y^7}{630}-\frac{559 y^9}{22680}-\frac{2651
   y^{11}}{113400}+O\left(y^{13}\right)$$ the remaining integral can be approximated integrating termwise.
Without any special function
$$x\sqrt{\cos (x)}=x-\frac{x^3}{4}-\frac{x^5}{96}-\frac{19 x^7}{5760}-\frac{559
   x^9}{645120}-\frac{29161 x^{11}}{116121600}+O\left(x^{13}\right)$$
$$\int x\sqrt{\cos (x)}\,dx=\frac{x^2}{2}-\frac{x^4}{16}-\frac{x^6}{576}-\frac{19 x^8}{46080}-\frac{559
   x^{10}}{6451200}-\frac{29161 x^{12}}{1393459200}+O\left(x^{14}\right)$$
A: Wolfram's calculation is ugly: https://www.wolframalpha.com/input/?i=integrate+x*sqrt%28cos%28x%29%29+dx
But to answer your question directly: yes it can be solved.
