I'm looking for a closed-form expression for the value of this integral:
$$I=\int_0^\pi \frac{\sin(x)}{\sqrt{x^3+x+1}} dx$$
The graph of the integrand looks like this:
$\hskip 2.4 in$
Numerically, the area is $0.875044...$ for which the Inverse Symbolic Calculator doesn't produce anything promising. My CAS finds neither an antiderivative nor a closed form for the definite integral, and my own manipulations haven't really got me anywhere either.