I have the following expression
$$A(x) = d \, {}_2 F_1(1/2 - i/2, 1; 3/2-i/2; 1-e^x) + \bar{d} \, {}_2 F_1(1/2 - i/2, 1; 3/2-i/2; 1-e^x), $$ where $d$ is a complex number and $\bar{d}$ its complex conjugate, $x$ is real and ${}_2 F_1(a,b;c;x)$ is the hypergeometric function.
This comes from an integral involving the product of logarithms, exponentials and cosines. Though numerically is checked that $A$ has no imaginary part, I wonder if I can further simplify things of the form
$$ A(x) = d \, _2F_1(a,b;c;x) + \bar{d} \, _2F_1(\bar{a},b;\bar{c};x) $$
I have looked up in Abramowitz & Stegun, but I fail to find such a simplifying property. Does anyone have a thought? I would appreciate it a lot.
Thanks in advance.