Integration shows different result after manual u-substitution

Question

I'm working with sine and cosine integral functions, and stumbled on this weird issue when performing integration in Mathematica.

I worked out the following integral by hand and found:

\begin{equation} \int_{-A}^{A} \frac{\sin(B(t-t_0)+Ct)}{t-t_0} dt = \\ \sin(Ct_0) \Big(\textrm{Ci}\big((B+C)(A-t_0)\big)-\textrm{Ci}\big(-(B+C)(A+t_0)\big)\Big) + \\cos(Ct_0) \Big( \textrm{Si}\big( (B+C)(A-t_0)\big) + \textrm{Si}\big((B+C)(A+t_0)\big) \Big), \end{equation}

with $\{A, B, C\} \in \Re_0$, and $\textrm{Si}, \textrm{Ci}$ denoting the Sine Integral and Cosine Integral function, respectively.

However, when I try to solve the same integral in Mathematica, I get a different solution:

Integrate[Sin[B (t - t0) + C t] / (t - t0), {t, -A, A}, GenerateConditions -> False] //FullSimplify
>(CosIntegral[-((B + C) (A - t0))] - CosIntegral[(B + C) (A + t0)]) Sin[C t0] + Cos[C t0] (SinIntegral[(B + C) (A - t0)] + SinIntegral[(B + C) (A + t0)])

where the argument of the first two Cosine Integral functions is suddenly negative compared to my solution by hand. I am fairly certain of my solution, so it would suprise me if I solved it incorrectly somehow.

Something I did try, was performing u-substitution by hand first, which results in:

$$\textrm{Let } u = B(t - t_0) + C t, \\\\$$

$$\int_{-A(B+C)-Bt_0}^{A(B+C)-Bt_0} \frac{\sin(u)}{u-t_0C} du$$

and solving this integral using Mathematica. This does give me my original solution again:

Integrate[Sin[u]/(u - t0*C), {u, -A (B + C) - B t0, A (B + C) - B t0}, GenerateConditions -> False] // FullSimplify
>(CosIntegral[(B + C) (A - t0)] - CosIntegral[-((B + C) (A + t0))]) Sin[C t0] + Cos[C t0] (SinIntegral[(B + C) (A - t0)] + SinIntegral[(B + C) (A + t0)])

So I don't really understand why Mathematica gives a different result before and after the u-substitution (assuming that my u-substitution is correct).

Am I missing something?

Answer 1

According to Mathematica's definition of the cosine integral function, we have $$ \text{Ci}(-x) = \text{Ci}(x) + \ln(-x) - \ln(x) = \text{Ci}(x) + i \pi $$ for $x > 0$. This then implies that $\text{Ci}(-x) - \text{Ci}(-y) = \text{Ci}(x) - \text{Ci}(y)$. In other words, the two forms of the integral are functionally equivalent so long as $x > 0$ and $y > 0$.

Mathematica can be induced to show this using FunctionExpand — though if I try to prove the above statement using Reduce, it returns False, which is strange. It's entirely possible that I'm missing something.

Reduce[
 {CosIntegral[x] - CosIntegral[y] == FunctionExpand[CosIntegral[-x] - CosIntegral[-y])], 
  x > 0, y > 0}, Reals]

(* False *)