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2

Simply add as an option Integrate[myExpression, {s, 0, z}, Assumptions -> z > 0] The reason why you got this error is because by default, Mathematica does not assume that the variable $z$ is a real number. It is trying to find an analytic solution to the integral for general complex $z$.

2

Perhaps this will help:- eqn = a Integrate[Exp[-s^2]/(s - c x)^2, {s, -Infinity, Infinity}, Assumptions -> Im[c x] != 0] + b Integrate[Exp[-s^2]/(s - d x)^2, {s, -Infinity, Infinity}, Assumptions -> Im[d x] != 0]; a = 1.5; b = 2; c = 0.8; d = 1; Plot[eqn, {x, -1000, 1000}]

6

The integral is conditionally convergent. You can progress using substitution: $u=2^{\frac{r}{b}}\iff r= b\log_2 u$ Hence,$\frac{dr}{du}=\frac{b}{u\ln 2}$ You can do these substitutions in Mathematica: f[r_, b_, la_, k_] := 2^(r/b) Exp[k (2^(r/b) - 1)/la]/(b la) exp = f[x, a1, a2, a3] /. {2^(x/a1) -> u}; ex = D[a1 Log[2, u], u]; ans = Integrate[a1 ...

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