galanakis
commented
3 years ago Below is the argument (complex number angle) of the antiderivative of the above function over the complex plain.
Antiderivative from exponentials: there is a branch-cut splitting the imaginary axis (the contour of integration) in half).
Antiderivative from trigonometric: there is no branch cut
One resolution would be to detect when branch-cuts partition the complex plane such that there is no integration contour that can avoid them, and remove their "height" from the result.
In this particular case, the anti-derivative is a[z] = -(1/2) ExpIntegralEi[-z] + ExpIntegralEi[z]/2 The contour is z=i t and the branch cut is at t=0.
Rubi returns the value a[i ∞] - a[-i ∞]. Instead it should return a[i ∞] - a[0+] + a[0-] - a[-i ∞] where a[0+], a[0-] are the limiting values right above and below the branch-cut respectively.
The correct value of this integral is π,
However Rubi gives 2π when the integrand has exponentials and π (correctly) when it is trigonometric.
The antiderivative with exponentials has ExpIntegralEi[z] and ExpIntegralEi[-z], which have branch cuts running in opposite directions (here z = i x). The antiderivative of the trigonometric form is SinIntegral without branch-cuts.
The problem is that the two branch-cuts cover the entire real axis and always cross the contour. I suppose that the branch cuts need to be moved so that the contour does not cross any of them.
In definite integrals, Rubi needs to detect the branch-cuts along the integration contour and "skip" them (or subtract their height). More below.