Integration $\int_0^1 \cos \left(\sqrt{c+ b x+x^2}\right) \, dx$

Question

How to integrate $\int_0^1 \cos \left(\sqrt{c+b x+x^2}\right) \, dx$? $b$ and $c$ can be any value (except both equal to zero) as long as the integration can be done.

Answer 1

As the integrand has two degrees of freedom, the integration bounds are virtually immaterial and the definite integral isn't simpler than the indefinite one.

And the indefinite one doesn't have a closed-form expression. By a linear change of variable, you can reduce to $$\int \cos\left(\sqrt{t^2+1}\right)\,dt$$ then $$\int \cos(\cosh u)\cosh u\,du$$ or $$\int\frac{u\cos(u)}{\sqrt{u^2-1}}\,du$$ but this leads you nowhere.