Integrate $(\sin x)^{\sin x}\,dx$

Question

Is there any way to calculate $$\int{(\sin x)^{\sin x}\,dx}$$

I do this steps:

Answer 1

Is there any way to calculate $\displaystyle\int{(\sin x)^{\sin x}\,dx}$ ?

No. See Liouville's theorem and the Risch algorithm.

Answer 2

Hint:

$\int(\sin x)^{\sin x}~dx$

$=\int(e^{\ln\sin x})^{\sin x}~dx$

$=\int e^{\sin x\ln\sin x}~dx$

$=\int\sum\limits_{n=0}^\infty\dfrac{\sin^nx(\ln\sin x)^n}{n!}dx$

$=\int\left(1+\sum\limits_{n=1}^\infty\dfrac{\sin^nx(\ln\sin x)^n}{n!}\right)dx$