FCIteratedIntegral[f,x,a,b] is a special head indicating
that the function f represents an
iterated integral or a linear combination thereof and that it should be
integrated in x from a to b. This
notation is understood by the function
FCIteratedIntegralEvaluate that does the actual
integration.
Notice that before applying FCIteratedIntegralEvaluate
all rational functions of x in f should be converted to the
FCPartialFractionFormrepresentation.
Overview, FCIteratedIntegralEvaluate, ToFCPartialFractionForm
fun = 1/(1 + x)\frac{1}{x+1}
int = FCIteratedIntegral[ToFCPartialFractionForm[fun, x], x, a, b]\text{FCIteratedIntegral}\left(\text{FCPartialFractionForm}\left(0,\left( \begin{array}{cc} \{x+1,-1\} & 1 \\ \end{array} \right),x\right),x,a,b\right)
FCIteratedIntegralEvaluate[int]G(-1; b)-G(-1; a)