Can this integeral be evaluated? $$\int_0^\infty\cos(ax)\sec(bx)\mathrm{d}x$$
Well, this problem came from the integral below: $$\int_0^\infty\cos(ax)\mathrm{sech}(bx)\mathrm{d}x$$ where $\mathrm{sech}(\cdot)$ is the hyperbolic secant function, i.e., $\mathrm{sech}(x)=\dfrac{2}{\mathrm{e}^x+\mathrm{e}^{-x}}$. I found the common solution is by Residual Theorem like this one. From my perspective, this may be resolved if there is a way to evaluate $\int_0^\infty\cos(ax)\sec(bx)\mathrm{d}x$ since $\mathrm{sech}(x)=\sec(\mathrm{i}x)$. At this point, the problem can be reduced to evaluate $\int_0^\infty\cos(ax)\sec(\mathrm{i}bx)\mathrm{d}x$.
So how can I evaluate $\int_0^\infty\cos(ax)\sec(bx)\mathrm{d}x$?
