Numerical evaluation of Goursat’s infinite integral

The infinite integral \(\int_0^{\infty}x\,dx/(1+x^6\sin^2x)\) converges but is hard to evaluate because the integrand f(x) = x/(1 + x6sin2x) is a non-convergent and unbounded function, indeed f(kπ) = kπ→ ∞ (k→ ∞). We present an efficient method to evaluate the above integral in high accuracy and actually obtain an approximate value in up to 73 significant digits on an octuple precision system in C++.