Question: How to integrate this case

1. I use your previous reply on V_G derive  on other characteristic function, but most are undefined

or complicated solution or can not evaluate, why?

for example

charc := 1+i*X/(i*X-1);
int(exp(-I*X*u)*charc, X = -infinity .. infinity);

it got this complicated thing
piecewise(Im(1/i) = 0, undefined, int(exp(-I*X*u)*(1+i*X/(i*X-1)), X = -infinity .. infinity, method = _UNEVAL))

Density := int(exp(-I*X*u)*charc, X = -infinity .. infinity);
Density := simplify(subs(X=g, Density));
tm := simplify(int(expand(1/(rho*sqrt(2*Pi*g))*exp(-((X-theta*g)^2)/(2*g*rho^2))*Density),g=0..infinity));
a := int(exp(I*X*u)*tm, X = -infinity .. infinity);
madan := simplify(a, power) assuming rho > 0, theta > 0, v > 0;
mean := subs(i=sqrt(-1), subs(t=0, simplify((i^(1))*diff(madan,t))));
var := subs(i=sqrt(-1), subs(t=0, simplify((i^(2))*diff(madan,t$2))));

2. second problem is

i use formula of weight function in Sturm-Liouville Theory and Orthogonal Functions

http://arxiv.org/abs/0906.3209

for example Hermit is success to get the generating function with weight function p

however for Gamma has difficulty

i near success but for Gamma, can not further evaluate the integration and get the same gamma density

restart;
GammaDensity := (b^a)*(x^(a-1))*exp(-x*b)/GAMMA(a);
restart;
with(SumTools):
with(inttrans):
b1:=a+c-x;
a1:=x;
p1 := subs(x=z, simplify(exp(int((b1-diff(a1,x))/a1,x))));
p2 := Summation(1/m!*(-b1*z)^m, m=0..infinity);
genfun := p1*p2;

After change genfun into characteristic function below

charc := exp(i*X)*exp(i*X*a)*exp(i*X*b)*i*X/((i*X)^a*(i*X)^b)
Density := simplify(int(exp(-I*X*u)*charc, X = -infinity .. infinity));
can not further evaluate to (b^a)*(x^(a-1))*exp(-x*b)/GAMMA(a)