# Question:Do you have any rules or approaches that could help me simplify this huge expression ?

## Question:Do you have any rules or approaches that could help me simplify this huge expression ?

Maple 13

I'm using maple 13.

The expression that I paste below is quite large and subsequent expressions are also swelling. This is causing maple to slow down and in some instances I get a message saying the kernel has disconnected (maple just times out) I want a way to programmatically reduce the size of the output. I have tried simplify(expr,size) and this has helped some. I don't want to use simplify(expr,symbolic) because of the presence of the square roots. Is there anything I can do to reduce its size ? Here beta and Q are real.

expr:=(1/2*((1/2592)*Q*((Q*Pi-1/2)^2*beta^2+Pi^2)^5*((((Q*Pi-1/2)^2*beta^2+Pi^2)*((((Q^2*Pi^2-1/2)*(1/2+Q*Pi)^2*(Q*Pi-1/2)*beta^5+(I*Pi^5*Q^4-(1/4*I)*Pi^3*Q^2+I*Pi^4*Q^3+(1/8*I)*Pi)*beta^4+((1/2)*Q^2*Pi^4+(1/2)*Pi^2+Q*Pi^3+3*Pi^5*Q^3)*beta^3+(3*I)*Pi^3*(1/3+Q^2*Pi^2)*beta^2+(2*Q*Pi^5+Pi^4)*beta+(2*I)*Pi^5)*cos(k*theta)+(1/2)*Q*Pi*beta^2*((1/2+Q*Pi)^2*(Q*Pi-1/2)*beta^3+I*Pi*(Q*Pi-3/2)*(1/2+Q*Pi)*beta^2+(Q*Pi^3+(3/2)*Pi^2)*beta+I*Pi^3))*exp(sigma^2*k^2)*exp(-(1/2)*sigma^2*k^2)-(1/2)*Q*Pi*beta^2*((1/2+Q*Pi)^2*(Q*Pi-1/2)*beta^3+I*Pi*(Q*Pi-3/2)*(1/2+Q*Pi)*beta^2+(Q*Pi^3+(3/2)*Pi^2)*beta+I*Pi^3)*exp(sigma^2*k^2)-((Q^2*Pi^2-1/2)*(1/2+Q*Pi)^2*(Q*Pi-1/2)*beta^5+(I*Pi^5*Q^4-(1/4*I)*Pi^3*Q^2+I*Pi^4*Q^3+(1/8*I)*Pi)*beta^4+((1/2)*Q^2*Pi^4+(1/2)*Pi^2+Q*Pi^3+3*Pi^5*Q^3)*beta^3+(3*I)*Pi^3*(1/3+Q^2*Pi^2)*beta^2+(2*Q*Pi^5+Pi^4)*beta+(2*I)*Pi^5)*cos(k*theta))*sqrt(((-1+4*Q^2*Pi^2)*beta^2+(8*I)*Pi^2*beta*Q-4*Pi^2)/(4*(Q*Pi-1/2)^2*beta^2+4*Pi^2))+(((1/2)*Q^2*Pi^2-1/8)*beta^2+I*Pi^2*beta*Q-(1/2)*Pi^2)*((((-(1/2)*Q^2*Pi^2+1/8)*beta^4+(2*Q^2*Pi^4+Pi^2)*beta^2+2*Pi^4)*cos(k*theta)+Q*((-1/4+Q^2*Pi^2)*beta^2+Pi^2)*Pi*beta^2)*exp(sigma^2*k^2)*exp(-(1/2)*sigma^2*k^2)-Q*((-1/4+Q^2*Pi^2)*beta^2+Pi^2)*Pi*beta^2*exp(sigma^2*k^2)-(2*((-(1/4)*Q^2*Pi^2+1/16)*beta^4+(Q^2*Pi^4+(1/2)*Pi^2)*beta^2+Pi^4))*cos(k*theta))*Pi)*Pi*sqrt(((-1+4*Q^2*Pi^2)*beta^2-(8*I)*Pi^2*beta*Q-4*Pi^2)/(4*(Q*Pi-1/2)^2*beta^2+4*Pi^2))+((2*(((1/2+Q*Pi)^4*(Q*Pi-1/2)^4*beta^8+((7/128)*Pi^2+4*Q^6*Pi^8-(9/8)*Q^4*Pi^6-(3/16)*Q^2*Pi^4)*beta^6+((13/2)*Pi^8*Q^4+(3/2)*Q^2*Pi^6+(11/32)*Pi^4)*beta^4+((9/8)*Pi^6+5*Pi^8*Q^2)*beta^2+(3/2)*Pi^8)*cos(k*theta)+(1/4*((-1/4+Q^2*Pi^2)*beta^2-Pi*beta+Pi^2))*Q*Pi^3*((-1/4+Q^2*Pi^2)*beta^2+Pi*beta+Pi^2)*beta^2))*exp(sigma^2*k^2)*exp(-(1/2)*sigma^2*k^2)-(1/2*((-1/4+Q^2*Pi^2)*beta^2-Pi*beta+Pi^2))*Q*Pi^3*((-1/4+Q^2*Pi^2)*beta^2+Pi*beta+Pi^2)*beta^2*exp(sigma^2*k^2)-(2*((1/2+Q*Pi)^4*(Q*Pi-1/2)^4*beta^8+((7/128)*Pi^2+4*Q^6*Pi^8-(9/8)*Q^4*Pi^6-(3/16)*Q^2*Pi^4)*beta^6+((13/2)*Pi^8*Q^4+(3/2)*Q^2*Pi^6+(11/32)*Pi^4)*beta^4+((9/8)*Pi^6+5*Pi^8*Q^2)*beta^2+(3/2)*Pi^8))*cos(k*theta))*sqrt(((-1+4*Q^2*Pi^2)*beta^2+(8*I)*Pi^2*beta*Q-4*Pi^2)/(4*(Q*Pi-1/2)^2*beta^2+4*Pi^2))+Pi*((((Q^2*Pi^2-1/2)*(1/2+Q*Pi)*(Q*Pi-1/2)^2*beta^5+(I*Pi^5*Q^4-(1/4*I)*Pi^3*Q^2-I*Pi^4*Q^3+(1/8*I)*Pi)*beta^4+(3*Pi^5*Q^3-(1/2)*Q^2*Pi^4-(1/2)*Pi^2+Q*Pi^3)*beta^3+(3*I)*Pi^3*(1/3+Q^2*Pi^2)*beta^2+(-Pi^4+2*Q*Pi^5)*beta+(2*I)*Pi^5)*cos(k*theta)+(1/2*((1/2+Q*Pi)*(Q*Pi-1/2)^2*beta^3+I*Pi*(Q*Pi+3/2)*(Q*Pi-1/2)*beta^2+(Q*Pi^3-(3/2)*Pi^2)*beta+I*Pi^3))*Q*Pi*beta^2)*exp(sigma^2*k^2)*exp(-(1/2)*sigma^2*k^2)-(1/2*((1/2+Q*Pi)*(Q*Pi-1/2)^2*beta^3+I*Pi*(Q*Pi+3/2)*(Q*Pi-1/2)*beta^2+(Q*Pi^3-(3/2)*Pi^2)*beta+I*Pi^3))*Q*Pi*beta^2*exp(sigma^2*k^2)-((Q^2*Pi^2-1/2)*(1/2+Q*Pi)*(Q*Pi-1/2)^2*beta^5+(I*Pi^5*Q^4-(1/4*I)*Pi^3*Q^2-I*Pi^4*Q^3+(1/8*I)*Pi)*beta^4+(3*Pi^5*Q^3-(1/2)*Q^2*Pi^4-(1/2)*Pi^2+Q*Pi^3)*beta^3+(3*I)*Pi^3*(1/3+Q^2*Pi^2)*beta^2+(-Pi^4+2*Q*Pi^5)*beta+(2*I)*Pi^5)*cos(k*theta))*((1/2+Q*Pi)^2*beta^2+Pi^2))*((1/2+Q*Pi)^2*beta^2+Pi^2)/(sqrt(((-1+4*Q^2*Pi^2)*beta^2+(8*I)*Pi^2*beta*Q-4*Pi^2)/(4*(Q*Pi-1/2)^2*beta^2+4*Pi^2))*(((1/2)*Q^2*Pi^2-1/8)*beta^2+I*Pi^2*beta*Q-(1/2)*Pi^2)*(-(1/3*(1/2+Q*Pi))*(Q*Pi-1/2)^2*beta^3+I*Pi*(Q*Pi+1/6)*(Q*Pi-1/2)*beta^2+(-(1/6)*Pi^2+Q*Pi^3)*beta-(1/3*I)*Pi^3)^2*exp(sigma^2*k^2)*beta^4*((1/3*(1/2+Q*Pi))*(Q*Pi-1/2)^2*beta^3+I*Pi*(Q*Pi+1/6)*(Q*Pi-1/2)*beta^2+((1/6)*Pi^2-Q*Pi^3)*beta-(1/3*I)*Pi^3)^2*((-(1/2)*Q^2*Pi^2+1/8)*beta^2+I*Pi^2*beta*Q+(1/2)*Pi^2)*((-Q*Pi+1/2)*beta+I*Pi)^2*((Q*Pi-1/2)*beta+I*Pi)^2)+65536*Q*((1+beta^2*Q^2)*Pi^2-beta^2*Q*Pi+(1/4)*beta^2)^5*(-1+exp(-(1/2)*sigma^2*k^2))*((2*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*beta^2*Q^2+Q*(sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+(4*I)*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+1)*beta-2*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+I*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+I)*Pi^2-(1/2)*beta*(-1+sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*Pi-(1/2)*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*beta^2)*((-(1+beta^2*Q^2)*(-2*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*beta^4*Q^4-4*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*beta^2*Q^2-Q*(sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*beta-I*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))-2*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+I*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*Pi^6+(1/2)*beta*(-4*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*beta^5*Q^5+2*Q^3*(-4*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))-I*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+I*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*beta^3+Q^2*(sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*beta^2-4*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*beta*Q-sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))-sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*Pi^5-(1/4)*beta^2*(2*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*beta^4*Q^4+Q^3*(sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*beta^3+Q^2*(-4*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))-I*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+I*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*beta^2-2*Q*(sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*beta-(2*I)*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))-6*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+(2*I)*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*Pi^4+(1/8)*beta^3*(8*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*beta^3*Q^3+Q^2*(sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*beta^2-8*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*beta*Q-2*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))-2*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*Pi^3-(1/16)*beta^4*(2*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*beta^2*Q^2-Q*(sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*beta-I*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+I*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))-6*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*Pi^2-(1/32)*beta^5*(sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+4*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*beta*Q+sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*Pi+(1/32)*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*beta^6)*cos(k*theta)+(1/16)*Pi^2*beta^2*Q*((8*(1+beta^2*Q^2))*(-Q*(sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*beta+I*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))-I*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*Pi^3+4*beta*(Q^2*(sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*beta^2+(2*I)*Q*(-sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*beta+3*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+3*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*Pi^2-2*beta^2*(-Q*(sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*beta+(3*I)*sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))-(3*I)*sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))*Pi-beta^3*(sqrt(-(4*Pi^2+beta^2-4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))+sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2)))))*((1+beta^2*Q^2)*Pi^2+beta^2*Q*Pi+(1/4)*beta^2)/(sqrt((-4*Pi^2-beta^2+4*beta^2*Q^2*Pi^2+(8*I)*Pi^2*beta*Q)/(4*Pi^2+beta^2-4*beta^2*Q*Pi+4*beta^2*Q^2*Pi^2))*((-8*beta^3*Q^3+(24*I)*beta^2*Q^2+24*beta*Q-8*I)*Pi^3+(-(8*I)*beta^2*Q-4*beta+4*beta^3*Q^2)*Pi^2+(-(2*I)*beta^2+2*beta^3*Q)*Pi-beta^3)^2*((4-4*beta^2*Q^2+(8*I)*beta*Q)*Pi^2+beta^2)*beta^4*((-2*beta*Q+2*I)*Pi+beta)^2*((8*beta^3*Q^3+(24*I)*beta^2*Q^2-24*beta*Q-8*I)*Pi^3+(-(8*I)*beta^2*Q+4*beta-4*beta^3*Q^2)*Pi^2+(-(2*I)*beta^2-2*beta^3*Q)*Pi+beta^3)^2*((2*beta*Q+2*I)*Pi-beta)^2*((-4+4*beta^2*Q^2+(8*I)*beta*Q)*Pi^2-beta^2))))*t^2;

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