FCIteratedIntegralEvaluate[ex] evaluates iterated integrals in ex in terms of multiple polylogarithms.
To that aim the ex must contain ration functions (in the FCPartialFractionForm notation) and possibly FCGPLs wrapped with FCIteratedIntegral heads
Overview, FCIteratedIntegral, FCIteratedIntegralSimplify, FCGPL.
int = FCPartialFractionForm[0, {{{-a + x[2], -1}, (1 + a + x[3])^(-2)},
{{1 + x[2] + x[3], -2}, -(1 + a + x[3])^(-1)}, {{1 + x[2] + x[3], -1}, -(1 + a + x[3])^(-2)}}, x[2]]\text{FCPartialFractionForm}\left(0,\left( \begin{array}{cc} \{x(2)-a,-1\} & \frac{1}{(a+x(3)+1)^2} \\ \{x(2)+x(3)+1,-2\} & -\frac{1}{a+x(3)+1} \\ \{x(2)+x(3)+1,-1\} & -\frac{1}{(a+x(3)+1)^2} \\ \end{array} \right),x(2)\right)
FCIteratedIntegralEvaluate[FCIteratedIntegral[int, x[2], 0, Infinity]]-\text{FCPartialFractionForm}\left(0,\left( \begin{array}{cc} \{\infty ,-1\} & -\frac{1}{a+x(3)+1} \\ \end{array} \right),\infty \right)+\text{FCPartialFractionForm}\left(0,\left( \begin{array}{cc} \{x(3)+1,-1\} & -\frac{1}{a+x(3)+1} \\ \end{array} \right),0\right)-\frac{G(-x[3]-1; \infty )}{(a+x(3)+1)^2}+\frac{G(a; \infty )}{(a+x(3)+1)^2}
FCIteratedIntegralEvaluate[FCIteratedIntegral[int, x[2], 0, x[2]]]\text{FCPartialFractionForm}\left(0,\left( \begin{array}{cc} \{x(3)+1,-1\} & -\frac{1}{a+x(3)+1} \\ \end{array} \right),0\right)-\text{FCPartialFractionForm}\left(0,\left( \begin{array}{cc} \{x(2)+x(3)+1,-1\} & -\frac{1}{a+x(3)+1} \\ \end{array} \right),x(2)\right)+\frac{G(a; x[2])}{(a+x(3)+1)^2}-\frac{G(-x[3]-1; x[2])}{(a+x(3)+1)^2}