On $\int_0^{\pi} \operatorname{Si}^n(x) \ \mathrm{d}x $ The Sine Integral, $\operatorname{Si}(x)$, is defined as $$\operatorname{Si}(x)=\int_0^x \frac{\sin t}{t} \ \mathrm{d}t $$
It follows that $$\int_0^{\pi} \operatorname{Si}(x) \ \mathrm{d}x = \bigg[x \operatorname{Si}(x) \bigg]_0^{\pi}-\int_0^{\pi} \sin x \ \mathrm{d}x = \pi \operatorname{Si} (\pi)-2$$ and further $$\int_0^{\pi} \operatorname{Si}^2 (x) \ \mathrm{d}x = \bigg[ x \operatorname{Si}^2(x) \bigg]_0^{\pi} -2\int_0^{\pi} \operatorname{Si} (x) \sin x \ \mathrm{d}x \\ = \pi \operatorname{Si}^2(\pi) -2 \left( \big[ -\cos x \operatorname{Si} (x) \big]_0^{\pi}+\int_0^{\pi} \frac{\sin x \cos x}{x} \ \mathrm{d}x \right)\\ = \pi \operatorname{Si}^2(\pi)-2 \operatorname{Si} (\pi)-\int_{2x=0}^{2\pi}\frac{\sin 2x}{2x} \ \mathrm{d}(2x)\\ = \pi\operatorname{Si}^2(\pi)-2 \operatorname{Si} (\pi)-\operatorname{Si} (2\pi)$$
Following a similar procedure on $\int_0^{\pi} \operatorname{Si}^3(x) \ \mathrm{d}x$ doesn’t quite work, because it requires evaluating the integral $\int_0^{\pi} \operatorname{Si}^2(x) \sin x \ \mathrm{d}x$, or equivalently, $\int_0^{\pi} \frac{\operatorname{Si}(x) \sin 2x}{x} \ \mathrm{d}x$.
This leads me to ask, can (and how can) $\int_0^{\pi} \operatorname{Si}^3(x) \ \mathrm{d}x$ be expressed in terms of elementary or special functions? How about the general case, i.e.  $$\color{purple}{\int_0^{\pi} \operatorname{Si}^n(x) \ \mathrm{d}x} \; ? $$
 A: Here is an observation that seems to hint a negative answer to OP's question.

Considering that $\pi$ does not play a particular role in the special values for $\operatorname{Si}(\cdot)$, the question would be almost equivalent to investigating the antiderivatives of $\operatorname{Si}(x)^k$'s. Now let us write
$$F(x) = \int_{x}^{\infty} \frac{\sin t}{t} \, \mathrm{d}t = \frac{\pi}{2} - \operatorname{Si}(x)$$
for the complementary version of $\operatorname{Si}(\cdot)$. In light of the binomial theorem, it is clear that $\int\operatorname{Si}(x)^n\,\mathrm{d}x$ can be computed if all of $\int F(x)^k\,\mathrm{d}x$ for $k=0,\dots,n$ can be computed. So we will shift our focus to finding the antiderivatives of $F(x)^k$'s.
Now employing OP's technique, we find that
\begin{align*}
\int F(x) \, \mathrm{d}x
&= x F(x) + \int \sin x \, \mathrm{d}x \\
&= x F(x) - \cos x + C
\end{align*}
and
\begin{align*}
\int F(x)^2 \, \mathrm{d}x
&= x F(x)^2 + 2\int F(x) \sin x \, \mathrm{d}x \\
&= x F(x)^2 - 2F(x)\cos x - \int \frac{2\sin x\cos x}{x} \, \mathrm{d}x \\
&= x F(x)^2 - 2F(x)\cos x + F(2x) + C.
\end{align*}
These computations, together with the asymptotic formula for $F(x)$, reveal the closed-forms for the following integrals:
$$ I_1 = \int_{0}^{\infty} F(x) \, \mathrm{d}x = 1 \qquad\text{and}\qquad I_2 = \int_{0}^{\infty} F(x)^2 \, \mathrm{d}x = \frac{\pi}{2}. $$
Similarly, if $\int F(x)^3 \, \mathrm{d}x$ admits a closed-form expression involving $F(\cdot)$, then we can expect that the integral
$$ I_3 = \int_{0}^{\infty} F(x)^3 \, \mathrm{d}x $$
will admit an elementary closed-form (or at least assume a simpler formula than $\int_{0}^{\pi}\operatorname{Si}(x)^3\,\mathrm{d}x$). However, we can prove that
$$I_3 = \frac{\pi^2}{4} - \frac{3}{2}\log^2 2 - \frac{3}{4}\operatorname{Li}_2\left(\frac{1}{4}\right) = 1.545982100082988\dots $$
holds, where $\operatorname{Li}_2(\cdot)$ is the dilogarithm. Since this expression seems non-elementary, I am skeptical about the ideal that $\operatorname{Si}(x)^3$ has an elementary antiderivative.

Addendum: Proof of the formula for $I_3$. We find that
\begin{align*}
\int F(x)^3 \, \mathrm{d}x
&= x F(x)^3 + 3\int F(x)^2 \sin x \, \mathrm{d}x \\
&= x F(x)^3 - 3F(x)^2\cos x - 6\int \frac{F(x)\sin x\cos x}{x} \, \mathrm{d}x \\
&= x F(x)^3 - 3F(x)^2\cos x - 3\int \frac{F(x)\sin(2x)}{x} \, \mathrm{d}x.
\end{align*}
From this, we get
$$ I_3 = \frac{3\pi^2}{4} - 3\int_{0}^{\infty} \frac{F(x)\sin(2x)}{x} \, \mathrm{d}x. $$
To compute this, we adopt the Feynman's trick. More precisely, we consider the parametrized integral
$$ I_3(s) = \frac{3\pi^2}{4} - 3\int_{0}^{\infty} \frac{F(x)\sin(sx)}{x} \, \mathrm{d}x. $$
Then
\begin{align*}
I_3'(s)
&= -3\int_{0}^{\infty} F(x)\cos(sx) \, \mathrm{d}x \\
&= -\frac{3}{s}\int_{0}^{\infty} \frac{\sin x \sin(sx)}{x} \, \mathrm{d}x \\
&= -\frac{3}{2s}\int_{0}^{\infty} \frac{\cos((s-1)x) - \cos((1+s)x)}{x} \, \mathrm{d}x
\end{align*}
The last integral can be computed using the Frullani's integral, yielding
\begin{align*}
I_3'(s)
&= -3\int_{0}^{\infty} F(x)\cos(sx) \, \mathrm{d}x \\
&= -\frac{3}{s}\int_{0}^{\infty} \frac{\sin x \sin(sx)}{x} \, \mathrm{d}x \\
&= \frac{3}{2s}(\log\left|s-1\right| - \log(s+1)).
\end{align*}
So it follows that
\begin{align*}
I_3 = I_3(2)
&= I_3(0) + \int_{0}^{2} I_3'(s) \, \mathrm{d}s \\
&= \frac{3\pi^2}{4}+ \int_{0}^{2} \frac{3}{2s}(\log\left|s-1\right| - \log(s+1)) \, \mathrm{d}s.
\end{align*}
By computing the last integral using the dilogarithm, we can complete the proof of the formula.
A: This note is divided into

*

*Part I: an elaboration of my comments (which have now been moved to chat), and


*Part II: in agreement with Sangchul's claim that an elementary closed form does not exist.
Part I: Writing the expression as a double integral, we have \begin{align}\int_0^\pi\frac{\operatorname{Si}(x)\sin2x}x\,dx&=\int_0^\pi\int_0^1\frac{\sin yx\sin2x}{yx}\,dy\,dx\\&=\frac12\int_0^1\int_0^\pi\frac{\cos((y-2)x)-\cos((y+2)x)}{yx}\,dx\,dy\\&=\frac12\int_0^1\frac{\operatorname{Ci}((y-2)\pi)-\log(y-2)-(\operatorname{Ci}((y+2)\pi)-\log(y+2))}y\,dy.\end{align} This follows from the series expansion $$\operatorname{Ci}(x)=\gamma+\log x+\sum_{n\ge1}\frac{(-1)^nx^{2n}}{2n(2n)!}\tag1$$ so that \begin{align}\lim_{x\to0}[\operatorname{Ci}((y+2)x)-\operatorname{Ci}((y-2)x)]&=\lim_{x\to0}\left[\log((y+2)x)-\log((y-2)x)+\sum_{n\ge1}\frac{f(n,y)x^{2n}}{2n(2n)!}\right]\\&=\log(y+2)-\log(y-2)\end{align} where \begin{align}f(n,y)&=(-1)^n((y+2)^{2n}-(y-2)^{2n})\\&=(-1)^n\sum_{0\le k\le2n}\binom{2n}ky^{2n-k}2^k(1-(-1)^k)\\&=(-1)^n\sum_{1\le k\le n}\binom{2n}{2k-1}y^{2n-2k+1}2^{2k}.\end{align} Using $(1)$ again, we obtain \begin{align}\int_0^\pi\frac{\operatorname{Si}(x)\sin2x}x\,dx&=-\frac12\int_0^1\sum_{n\ge1}\frac{(-1)^n\pi^{2n}f(n,y)}{2n(2n)!y}\,dy\\&=-\frac14\sum_{n\ge1}\sum_{1\le k\le n}\int_0^1\frac{(-1)^n\pi^{2n}4^k}{n(2k-1)!(2n-2k+1)!}y^{2n-2k}\,dy\\&=-\frac14\sum_{n\ge1}\sum_{1\le k\le n}\frac{(-1)^n\pi^{2n}4^k}{n(2n-2k+1)(2k-1)!(2n-2k+1)!}\end{align} where the sum-integral interchange is justified by Fubini. This can be rewritten in one summation as $$\int_0^\pi\frac{\operatorname{Si}(x)\sin2x}x\,dx=-2\sum_{n\ge1}\frac{(-1)^n\pi^{2n}}{(2n)!(2n-1)}{}_3F_2\left(\frac12-n,\frac12-n,1-n;\frac32,\frac32-n;4\right)$$ by expressing the denominator in terms of Pochhammer symbols. Here is verification.
Part 2: It is important to note that the upper limit of $\pi$ does not simplify the integral, as mentioned in Sangchul's answer. From Part I, we have \begin{align}\int_0^\pi\frac{\operatorname{Si}(x)\sin2x}x\,dx=\lim_{\epsilon\to0^+}\int_\epsilon^1\frac{\operatorname{Ci}((y-2)\pi)-\log(y-2)-(\operatorname{Ci}((y+2)\pi)-\log(y+2))}{2y}\,dy.\end{align} The antiderivatives of $\log(y\pm2)/y$ can be readily expressed in terms of dilogarithms, so the only issue stems from the evaluation of $$\int_\epsilon^1\frac{\operatorname{Ci}((y\pm2)\pi)}y\,dy=\int_{(\epsilon\pm2)\pi}^{(1\pm2)\pi}\frac{\operatorname{Ci}(w)}{w\mp2\pi}\,dw.$$ This is why I strongly doubt there is an elementary closed form for $\int_0^\pi\operatorname{Si}^3(x)\,dx$, as $\operatorname{Ci}$ is non-periodic and the limits do not allow for any simplifications to be made.
A: I am not sure my result is entirely correct or not but if there is any mistakes then please tell me in the comments.
So the integral is, $\textstyle\displaystyle{\int_{0}^{\pi}\operatorname{Si}^n(x)dx}$
I think we can start to simplify this using integration by parts, more specifically the DI method
(If you don't know what DI method is then for introduction you can watch this and this video.

So let's start-
$\begin{array}
\ & \text{D} & \text{I}\\
+ & \operatorname{Si}^n(x) & 1\\
- & n\operatorname{Si}^{n-1}(x)\frac{\sin(x)}{x} & x
\end{array}$
$\textstyle\displaystyle{\therefore\int_{0}^{\pi}\operatorname{Si}^n(x)dx}$
$\textstyle\displaystyle{=x\operatorname{Si}^n(x)\bigg]_{0}^{\pi}-n\int_{0}^{\pi}\sin(x)\operatorname{Si}^{n-1}(x)dx}$
$\textstyle\displaystyle{=\pi\operatorname{Si}^n(\pi)-n\int_{0}^{\pi}\sin(x)\operatorname{Si}^{n-1}(x)dx}$

Let's do this process again to find some pattern.
$\begin{array}
\ & \text{D} & \text{I}\\
+ & \sin(x)\operatorname{Si}^{n-1}(x) & 1\\
- & \begin{align}\cos(x)\operatorname{Si}^{n-1}(x) & +(n-1)\operatorname{Si}^{n-2}(x)\frac{\sin^2(x)}{x}\end{align} & x
\end{array}$
$\textstyle\displaystyle{\therefore\int_{0}^{\pi}\operatorname{Si}^n(x)dx}$
$\textstyle\displaystyle{=\pi\operatorname{Si}^n(\pi)-nx\operatorname{Si}^{n-1}(x)\sin(x)\bigg]_{0}^{\pi}+n\int_{0}^{\pi}x\cos(x)\operatorname{Si}^{n-1}(x)dx+n(n-1)\int_{0}^{\pi}\sin^2(x)\operatorname{Si}^{n-2}(x)dx}$
Here, $\textstyle\displaystyle{nx\operatorname{Si}^{n-1}(x)\sin(x)\bigg]_{0}^{\pi}=0}$

And let's do it for the last time
$\begin{array}
\ & \text{D} &\text{I}\\
+ & \sin^2(x)\operatorname{Si}^{n-2}(x) & 1\\
- & \begin{align}2\cos(x)\operatorname{Si}^{n-2}(x) & +(n-2)\operatorname{Si}^{n-3}(x)\frac{\sin^3(x)}{x}\end{align} & x
\end{array}$
$\textstyle\displaystyle{\therefore\int_{0}^{\pi}\operatorname{Si}^n(x)}$
$\textstyle\displaystyle{=\pi\operatorname{Si}^n(\pi)
+n\int_{0}^{\pi}x\cos(x)\operatorname{Si}^{n-1}(x)dx+n(n-1)x\sin^2(x)\operatorname{Si}^{n-2}(x)\bigg]_{0}^{\pi}-2n(n-1)\int_{0}^{\pi}x\cos(x)\operatorname{Si}^{n-2}(x)dx+n(n-1)(n-2)\int_{0}^{\pi}\sin^3(x)\operatorname{Si}^{n-3}(x)dx}$
Here, $\textstyle\displaystyle{n(n-1)x\sin^2(x)\operatorname{Si}^{n-2}(x)\bigg]_{0}^{\pi}=0}$

So now let's find patterns, at each step:-
$\pi\operatorname{Si}^n(\pi)$ stays, the term which is in the form
$\textstyle\displaystyle{\int_{0}^{\pi}\sin^k(x)\operatorname{Si}^{n-k}(x)dx}$ is broken into
$\textstyle\displaystyle{\frac{n!}{(n-k)!}x\sin^k(x)\operatorname{Si}^{n-k}(x)\bigg]_{0}^{\pi}}$,
$\textstyle\displaystyle{\frac{kn!}{(n-k)!}\int_{0}^{\pi}x\cos(x)\operatorname{Si}^{n-k}(x)dx}$ and
$\textstyle\displaystyle{\frac{n!}{[n-(k+1)]!}\int_{0}^{\pi}\sin^{k+1}(x)\operatorname{Si}^{n-(k+1)}(x)dx}$ where
$\textstyle\displaystyle{\frac{n!}{(n-k)!}x\sin^k(x)\operatorname{Si}^{n-k}(x)\bigg]_{0}^{\pi}}$ vanishes to $0$,
$\textstyle\displaystyle{\frac{n!k}{(n-k)!}\int_{0}^{\pi}x\cos(x)\operatorname{Si}^{n-k}(x)dx}$ stays and
$\textstyle\displaystyle{\frac{n!}{[n-(k+1)]!}\int_{0}^{\pi}\sin^{k+1}(x)\operatorname{Si}^{n-(k+1)}(x)dx}$ is broken down further.

So finally we get,
$\textstyle\displaystyle{\int_{0}^{\pi}\operatorname{Si}^n(x)dx}$
$\textstyle\displaystyle{=\pi\operatorname{Si}^n(\pi)$
 $+\sum_{k=1}^{n}\frac{n!k}{(n-k)!}\int_{0}^{\pi}x\cos(x)\operatorname{Si}^{n-k}(x)dx}$
This is as far as I have come. If I can work out more then I would edit my answer to add that.
