I would like to know if there is a closed formula for the integral $$ \int \cos(x)^{{\cos(x)}}\mathrm{d}x\overset{? }{=} $$ in terms of $\pm x^{\pm n}$, $(\pm\cos(x))^{\pm 1}$, $(\pm\sin(x))^{\pm 1}$, $ e^{\pm x}$, $ e^{\pm\cos(x)}$,$ e^{\pm\sin(x)}$ and Gamma function $\Gamma(x)$. For exemple $$ \int \cos(x)^{{\cos(x)}}\mathrm{d}x\overset{? }{=}{e^x\cdot \cos(x)}+{\Gamma(x)\cdot e^{2x}} $$ By 'closed formula' I mean if there is a polynomial in several variables $$ p(x_1,x_2,\ldots,x_r)=\sum_{i_1}\sum_{i_2}\ldots\sum_{i_r}a_{i_1,i_2,\ldots,i_r}x_1^{i_1}x_2^{i_2}\cdot\cdots\cdot x_r^{i_r} $$ such that $$ \int \cos(x)^{{\cos(x)}}\mathrm{d}x=p(\varphi_1(x),\varphi_2(x),\ldots,\varphi_r(x))+C $$ for $\varphi_1(x),\varphi_1(x),\ldots \varphi_r(x)\in \{\pm x^{\pm n}, (\pm\cos(x))^{\pm 1}, (\pm\sin(x))^{\pm 1}, e^{\pm x}, e^{\pm\cos(x)}, e^{\pm\sin(x)},\Gamma(x)^{\pm 1}\}$.
My attempt $$ \int \cos(x)^{{\cos(x)}}\mathrm{d}x = \int \exp\circ(\cos(x)\log({{\cos(x)}}))\mathrm{d}x = \int \exp\circ(y\log({{y}}))\sqrt{1-y}\mathrm{d}y $$
