How to evaluate the integral of $\int\sin(\sin(x))dx$? I recently watched a video on YouTube talking about something called as horseshoe integrals. A bit of research told me that that's not really a thing. Nonetheless, the YouTuber referred to the integral $$ \int\sin(\sin(x))dx $$ to be an example of a horseshoe integral and I haven't been able to come to a solution. Any leads would help.

$\int \sin (\sin (x))dx $ cannot be expressed in the form of elementary functions. If you still want to integrate it, you can do so using the infinite series of $\sin(x)$.

$$\sin(x)= \sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1} $$

$$\int \sin (\sin (x))dx=\int \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}(\sin x)^{2n+1} dx$$ Since $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}$ converges to $\sin(1)$ we can swap the integral sign with the summation.

$$\int \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}(\sin x)^{2n+1} dx= \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \int(\sin x)^{2n+1} dx$$

$$ \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \int(\sin x)^{2n+1} dx=$$ $$-\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}\cos(x)\sin^{2n}(x)sin^{2}(x)^{-n} 2F_1(1/2, -n; 3/2 ; cos^2(x))+C$$

where, $_2F_1 (a,b ;c;d)$ is the hyper geometric function.