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}