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I don't know if this is what you expected but maxima can integrate the entropy

sage:maxima_console()

(%i1) dispay2d:false;
(%o1)                                false
(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$(%i3) assume(sigma>0)$

(%i4) assume(mu>0)$(%i5) ii:integrate(-q*log(q),x,minf,inf)$

(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$(%i7) ev(%,integrate); 5/2 3/2 3/2 (%o7) (sqrt(%pi) (2 sigma log(sigma) + (2 log(%pi) + 2 log(2)) sigma) 3/2 5/2 + 2 sqrt(%pi) sigma)/(2 sqrt(%pi) sigma)  I don't know if this is what you expected but maxima can integrate the entropy sage:maxima_console() (%i1) dispay2d:false; (%o1) false (%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$

(%i3) assume(sigma>0)$(%i4) assume(mu>0)$

(%i5) ii:integrate(-q*log(q),x,minf,inf)$(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$

(%i7) ev(%,integrate);
5/2                      3/2             3/2
(%o7) (sqrt(%pi) (2    sigma log(sigma) + (2    log(%pi) + 2    log(2)) sigma)
3/2                    5/2
+ 2    sqrt(%pi) sigma)/(2    sqrt(%pi) sigma)


If ex is sageized version of the last expression then

sage: ex.full_simplify()
1/2*log(pi) + 1/2*log(2) + log(sigma) + 1/2


I don't know if this is what you expected but maxima can integrate compute the entropy

sage:maxima_console()

(%i1) dispay2d:false;
(%o1)                                false
(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$(%i3) assume(sigma>0)$

(%i4) assume(mu>0)$(%i5) ii:integrate(-q*log(q),x,minf,inf)$

(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$(%i7) ev(%,integrate); 5/2 3/2 3/2 (%o7) (sqrt(%pi) (2 sigma log(sigma) + (2 log(%pi) + 2 log(2)) sigma) 3/2 5/2 + 2 sqrt(%pi) sigma)/(2 sqrt(%pi) sigma)  If ex is sageized version of the last expression then sage: ex.full_simplify() 1/2*log(pi) + 1/2*log(2) + log(sigma) + 1/2  I don't know if this is what you expected but maxima can compute the entropy sage:maxima_console() (%i1) dispay2d:false; (%o1) false (%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$

(%i3) assume(sigma>0)$(%i4) assume(mu>0)$

(%i5) ii:integrate(-q*log(q),x,minf,inf)$(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$

(%i7) ev(%,integrate);
5/2                      3/2             3/2
(%o7) (sqrt(%pi) (2    sigma log(sigma) + (2    log(%pi) + 2    log(2)) sigma)
3/2                    5/2
+ 2    sqrt(%pi) sigma)/(2    sqrt(%pi) sigma)


If ex is sageized version of the last expression then

sage: ex.full_simplify()
1/2*log(pi) + 1/2*log(2) + log(sigma) + 1/2


and the last expression is equal: log(sqrt(2pi)sigma)+1/2

I don't know if this is what you expected but maxima can compute the entropy

sage:maxima_console()

(%i1) dispay2d:false;
(%o1)                                false
(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$(%i3) assume(sigma>0)$

(%i4) assume(mu>0)$(%i5) ii:integrate(-q*log(q),x,minf,inf)$

(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$(%i7) ev(%,integrate); 5/2 3/2 3/2 (%o7) (sqrt(%pi) (2 sigma log(sigma) + (2 log(%pi) + 2 log(2)) sigma) 3/2 5/2 + 2 sqrt(%pi) sigma)/(2 sqrt(%pi) sigma)  If ex is sageized version of the last expression then sage: ex.full_simplify() 1/2*log(pi) + 1/2*log(2) + log(sigma) + 1/2  and the last expression is equal: log(sqrt(2pi)sigma)+1/2 log(sqrt(2*pi)*sigma)+1/2 I don't know if this is what you expected but maxima can compute the entropy sage:maxima_console() (%i1) dispay2d:false; (%o1) false (%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$

(%i3) assume(sigma>0)$(%i4) assume(mu>0)$

(%i5) ii:integrate(-q*log(q),x,minf,inf)$(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$

(%i7) ev(%,integrate);
5/2 ev(%,integrate)$(%i8) ratsimp(%); 3/2 2 log(sigma) + log(%pi) + log(2) + 1 (%o8) 3/2 (%o7) (sqrt(%pi) (2 sigma log(sigma) + (2 log(%pi) + 2 log(2)) sigma) ------------------------------------ 3/2 5/2 + 2 sqrt(%pi) sigma)/(2 sqrt(%pi) sigma)  If ex is sageized version of the last expression then sage: ex.full_simplify() 1/2*log(pi) + 1/2*log(2) + log(sigma) + 1/2 2  and the last expression is equal: log(sqrt(2*pi)*sigma)+1/2 I don't know if this is what you expected but maxima can compute the entropy sage:maxima_console() (%i1) dispay2d:false; (%o1) false (%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$

(%i3) assume(sigma>0)$(%i4) assume(mu>0)$

(%i5) ii:integrate(-q*log(q),x,minf,inf)$(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$

(%i7) ev(%,integrate)$(%i8) ratsimp(%); 2 log(sigma) + log(%pi) + log(2) + 1 (%o8) ------------------------------------ 2 (%i9) logcontract(%); 2 log(2 %pi sigma ) + 1 (%o9) --------------------- 2  and the last expression is equal: log(sqrt(2*pi)*sigma)+1/2 I don't know if this is what you expected but maxima Maxima can compute the entropy sage:maxima_console() (%i1) dispay2d:false; (%o1) false (%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$

(%i3) assume(sigma>0)$(%i4) assume(mu>0)$

(%i5) ii:integrate(-q*log(q),x,minf,inf)$(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$

(%i7) ev(%,integrate)$(%i8) ratsimp(%); 2 log(sigma) + log(%pi) + log(2) + 1 (%o8) ------------------------------------ display2d:false$  2

(%i2) q:1/((2*%pi)^(1/2)*sigma)*exp(-(x - mu)^2/(2*sigma^2))$(%i3) assume(sigma>0)$

(%i4) assume(mu>0)$(%i5) ii:integrate(-q*log(q),x,minf,inf)$

(%i6) changevar(ii,(x-mu)/(sqrt(2)*sigma)-y,y,x)$(%i7) ev(%,integrate)$

(%i8) ratsimp(%);

(%o8) (2*log(sigma)+log(%pi)+log(2)+1)/2
(%i9) logcontract(%);

2
log(2 %pi sigma ) + 1
(%o9)                        ---------------------
2
(log(2*%pi*sigma^2)+1)/2


and the last expression is equal: equal to:

log(sqrt(2*pi)*sigma)+1/2