Try
(%i3) f(x):=%e^((-x^2)/(4*%sigma^2))/(2^(1/4)*%pi^(1/4)*sqrt(%sigma))$
(%i4) integrate(x*f(x^2-4),x,minf,inf);
(eq4) 0
(%i5) integrate(x^2*f(x^3-4),x,minf,inf);
(eq5) (2*%pi^(1/4)*sqrt(%sigma))/(3*2^(1/4))
(%i6) integrate((2*x-1)*f(x^5),x,minf,inf);
(eq6) -gamma(1/10)/(5*2^(1/20)*%pi^(1/4)*%sigma^(3/10))
I think this is right above but when the exponent is n you get an unending sequence of questions as long as you don't
answer no to any of them.
(%i17) integrate((2*x-1)*f(x^n),x,minf,inf);
Is 2*n an integer?y;
Is 2*n an even number?y;
Is 2*n an even number?y;
Is 2*n+1 an even number?y;
Is 2*n+1 an even number?y;
Is n-1 positive, negative, or zero?p;
Is 2*n-2 an even number?y;
Is 2*n-2 an even number?y;
Is 2*n-1 an even number?y;
Is 2*n-1 an even number?y;
Is 2*n an even number?y;
Is 2*n an even number?y;
Is 2*n+1 an even number?y;
Is 2*n+1 an even number?y;
Is 2*n-1 an even number?y;
Is 2*n-1 an even number?y;
Is n an integer?