Is it possible that this expression has an elementary one (specifically the dilog's):

Y0:=(1/16)*(s*t*(exp(2*t)*s+exp(4*t)+1)*ln((exp(2*t)*s-(-s^2+1)^(1/2)+1)^16*(1+(-s^2+1)^(1/2))^16/((exp(2*t)*s+(-s^2+1)^(1/2)+1)^16*(1-(-s^2+1)^(1/2))^16))+s^3*t*(exp(4*t)+1)*ln((exp(2*t)*s-(-s^2+1)^(1/2)+1)^8*(1+(-s^2+1)^(1/2))^8/((exp(2*t)*s+(-s^2+1)^(1/2)+1)^8*(1-(-s^2+1)^(1/2))^8))+exp(2*t)*t*ln((exp(2*t)*s-(-s^2+1)^(1/2)+1)^32*(1+(-s^2+1)^(1/2))^32/((exp(2*t)*s+(-s^2+1)^(1/2)+1)^32*(1-(-s^2+1)^(1/2))^32))+4*((exp(4*t)+1)*s+2*exp(2*t))*(s^2+2)*dilog((-exp(2*t)*s+(-s^2+1)^(1/2)-1)/(-1+(-s^2+1)^(1/2)))-4*((exp(4*t)+1)*s+2*exp(2*t))*(s^2+2)*dilog((exp(2*t)*s+(-s^2+1)^(1/2)+1)/(1+(-s^2+1)^(1/2)))+((32*s^2*t+64*t)*exp(2*t)+16*(((t+1/8)*s^2+2*t+2)*exp(4*t)-(5/4)*s*exp(-2*t)-(1/8)*exp(-4*t)*s^2+(5/4)*s*exp(6*t)+(1/8)*s^2*exp(8*t)+(t-1/8)*s^2-2+2*t)*s)*arctanh((exp(2*t)-1)*(-1+s)/((-s^2+1)^(1/2)*(exp(2*t)+1)))+8*(-s^2+1)^(1/2)*((1/8)*s*(exp(4*t)+1)*ln((exp(4*t)*s+2*exp(2*t)+s)^12/s^12)+(1/8)*exp(2*t)*ln((exp(4*t)*s+2*exp(2*t)+s)^24/s^24)+(s^2-6*t-3)*exp(2*t)+((-(1/8)*s^2-3*t)*exp(4*t)+s*exp(-2*t)+(1/8)*exp(-4*t)*s^2+s*exp(6*t)+(1/8)*s^2*exp(8*t)-(1/8)*s^2-3*t)*s))/((s*exp(-2*t)+exp(2*t)*s+2)*(exp(4*t)*s+2*exp(2*t)+s)*((-s^2+1)^(1/2)+2*arctanh((-1+s)/(-s^2+1)^(1/2))))

Also I'm wondering since Y0 should solve the ode

-(diff(diff(y(t), t), t))+(4-12/(1+s*cosh(2*t))+8*(-s^2+1)/(1+s*cosh(2*t))^2)*y(t) = C/(1+s*cosh(2*t))

with some constant C but I only get rubbish.

I ask this because I found that in another context this seems to be correct:

f1:=-(1/12)*Pi^2*((-s^2+1)^(1/2)-arccosh(1/s))/(-s^2+1)^(3/2)+(1/12)*arccosh(1/s)^3/(-s^2+1)^(3/2)-(1/4)*arccosh(1/s)^2/(-s^2+1)

f2:=(1/2)*((-s^2+1)^(1/2)*(polylog(2, s/(-1+(-s^2+1)^(1/2)))+polylog(2, -s/(1+(-s^2+1)^(1/2))))-polylog(3, s/(-1+(-s^2+1)^(1/2)))+polylog(3, -s/(1+(-s^2+1)^(1/2))))/(-s^2+1)^(3/2)

and f1=f2

but maple doesnt convert it.

Also maple has trouble to convert

2*arctanh(sqrt((1-s)/(1+s)))=arccosh(1/s)

everywhere: 0<s<1

How can I get maple to integrate this expression numerically.

For a specific value 0<s<1 it should be enough to integrate from -40..40 instead of -infinity..infinity

Anyway. My maple version always hangs up :-(

(1/2)*(-4*dilog(-(exp(2*t)*s-(-s^2+1)^(1/2)+1)/(-1+(-s^2+1)^(1/2)))*exp(4*t)+arctanh((-1+s)/(-s^2+1)^(1/2))*s^2+arctanh((exp(2*t)*s-exp(2*t)-s+1)/((exp(2*t)+1)*(-s^2+1)^(1/2)))*s^2+8*(-s^2+1)^(1/2)*exp(4*t)+4*dilog((exp(2*t)*s+(-s^2+1)^(1/2)+1)/(1+(-s^2+1)^(1/2)))*exp(4*t)+4*exp(4*t)*arctanh((-1+s)/(-s^2+1)^(1/2))-8*arctanh((exp(2*t)*s-exp(2*t)-s+1)/((exp(2*t)+1)*(-s^2+1)^(1/2)))*exp(4*t)*s^2*t-4*ln(1+(-s^2+1)^(1/2))*exp(4*t)*s^2*t+4*ln(1-(-s^2+1)^(1/2))*exp(4*t)*s^2*t-4*ln(exp(2*t)*s-(-s^2+1)^(1/2)+1)*exp(4*t)*s^2*t+4*ln(exp(2*t)*s+(-s^2+1)^(1/2)+1)*exp(4*t)*s^2*t+12*(-s^2+1)^(1/2)*exp(4*t)*t-16*arctanh((exp(2*t)*s-exp(2*t)-s+1)/((exp(2*t)+1)*(-s^2+1)^(1/2)))*exp(4*t)*t-8*ln(1+(-s^2+1)^(1/2))*exp(4*t)*t+8*ln(1-(-s^2+1)^(1/2))*exp(4*t)*t-8*ln(exp(2*t)*s-(-s^2+1)^(1/2)+1)*exp(4*t)*t+8*ln(exp(2*t)*s+(-s^2+1)^(1/2)+1)*exp(4*t)*t-(-s^2+1)^(1/2)*exp(2*t)*s+8*arctanh((exp(2*t)*s-exp(2*t)-s+1)/((exp(2*t)+1)*(-s^2+1)^(1/2)))*exp(2*t)*s+4*exp(2*t)*arctanh((-1+s)/(-s^2+1)^(1/2))*s-(-s^2+1)^(1/2)*exp(6*t)*s-8*arctanh((exp(2*t)*s-exp(2*t)-s+1)/((exp(2*t)+1)*(-s^2+1)^(1/2)))*exp(6*t)*s+4*exp(6*t)*arctanh((-1+s)/(-s^2+1)^(1/2))*s+2*dilog((exp(2*t)*s+(-s^2+1)^(1/2)+1)/(1+(-s^2+1)^(1/2)))*exp(4*t)*s^2+2*(-s^2+1)^(1/2)*exp(4*t)*s^2-arctanh((exp(2*t)*s-exp(2*t)-s+1)/((exp(2*t)+1)*(-s^2+1)^(1/2)))*exp(8*t)*s^2+exp(8*t)*arctanh((-1+s)/(-s^2+1)^(1/2))*s^2+2*exp(4*t)*arctanh((-1+s)/(-s^2+1)^(1/2))*s^2-6*(-s^2+1)^(1/2)*ln(exp(4*t)*s+2*exp(2*t)+s)*exp(4*t)+6*(-s^2+1)^(1/2)*ln(s)*exp(4*t)-2*dilog(-(exp(2*t)*s-(-s^2+1)^(1/2)+1)/(-1+(-s^2+1)^(1/2)))*exp(4*t)*s^2)/((-s^2+1)^(1/2)*exp(8*t)*s^2-2*arctanh((-s^2+1)^(1/2)/(1+s))*exp(8*t)*s^2+4*(-s^2+1)^(1/2)*exp(6*t)*s-8*arctanh((-s^2+1)^(1/2)/(1+s))*exp(6*t)*s+2*(-s^2+1)^(1/2)*exp(4*t)*s^2-4*arctanh((-s^2+1)^(1/2)/(1+s))*exp(4*t)*s^2+4*(-s^2+1)^(1/2)*exp(4*t)-8*arctanh((-s^2+1)^(1/2)/(1+s))*exp(4*t)+4*(-s^2+1)^(1/2)*exp(2*t)*s-8*arctanh((-s^2+1)^(1/2)/(1+s))*exp(2*t)*s+(-s^2+1)^(1/2)*s^2-2*arctanh((-s^2+1)^(1/2)/(1+s))*s^2)

Has anyone an idea how to integrate

int(t^2/(1+s*cosh(2*t))^2,t=-infinity..infinity)

0<s<1

I have following expression:

y1:=t->1/(4*cosh(t)^2)

I:=int(y1(t)^2,t=-T/2..T/2)

Now I tried:

MultiSeries:-asympt(I,T,5)

for which I only get the highest order.

Can I increase the order in any way?

I have following expression

f:=t->((1/8)*s^2*sinh(4*t)+t+(1/2)*s^2*t+s*sinh(2*t))/(1+s*cosh(2*t))

which is 1 solution of the ODE

ode2 := -(diff(y(t), t, t))+(4-12/(1+s*cosh(2*t))+(8*(-s^2+1))/(1+s*cosh(2*t))^2)*y(t) = 0

Now I wanted to construct 2 linear independent solutions via:

f1:=f(t_b-t)

f2:=f(t-t_a)

and calculate the Wronskian:

with(LinearAlgebra); with(VectorCalculus)

Determinant(Wronskian([f(t_b-t), f(t-t_a)], t))

Since I know these functions are solutions of the second order ODE which does not contain any first order derivative the Wronskian should be a constant. Unfortunately Maple has a hard time to simplify it since the epxression is a little big. Is it my fault or has anyone an idea what to do?