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here tau_f=-4/3 constant. The first part is an amplitude, the osc1 satisfies the linear ODE given above. I think this has Levin Potential if I can work out how to input the LevinRules given the above ODE? (Here and in the above I fix my parameters the ODE depends on to (1/10,1) to simplify giving the ICs but I don't that detracts from the main problem)

here tau_f constant. The first part is an amplitude, the osc1 satisfies the linear ODE given above. I think this has Levin potential if I can work out how to input the LevinRules given the above second order ODE? (Here and in the above I fix my parameters the ODE depends on to (1/10,1) to simplify giving the ICs but I don't that detracts from the main problem). Would need to work out what the Kernal is from the ODE above.

Exp[-I s] (ubar[tau_f - s])^(-i 4/10)
 Exp[+i 1/10 rstar[
tau_f - s]] Conjugate[osc1[rtau[tauf-s]]]

here tau_f=-4/3 constant. M=1. The first part is an amplitude, the osc1 satisfies the linear ODE given above. I think this has Levin Potential if I can work out how to input the LevinRules given the above ODE? (Here and in the above I fix my parameters the ODE depends on to (1/10,1) to simplify giving the ICs but I don't that detracts from the main problem)

Exp[-I s] (ubar[tau_f - s])^(-i 4/10)Exp[+i 1/10 rstar[tau_f - s]]osc1[rtau[tauf-s]]*

here tau_f=-4/3 constant. The first part is an amplitude, the osc1 satisfies the linear ODE given above. I think this has Levin Potential if I can work out how to input the LevinRules given the above ODE? (Here and in the above I fix my parameters the ODE depends on to (1/10,1) to simplify giving the ICs but I don't that detracts from the main problem)

UPDATE

Let's say I managed to split my integral into a few different integrals. First give the definitions:

vbar[tau_?
NumericQ] := (4 M) ((tau/tauh)^(1/3) + 1) Exp[-(tau/tauh)^(1/3) + 
 1/2 (tau/tauh)^(2/3) - 1/3 (tau/tauh)];
ubar[tau_?
NumericQ] := -(4 M) ((tau/tauh)^(1/3) - 1) Exp[(tau/tauh)^(1/3) + 
 1/2 (tau/tauh)^(2/3) + 1/3 (tau/tauh)];
rtau[tau_?NumericQ] := (2 M) (tau/tauh)^(2/3);

in addition to those made above, then I think I can give my integral as a sum of integrands that look like this

Exp[-I s] (ubar[tau_f - s])^(-i 4/10)
Exp[+i 1/10 rstar[
tau_f - s]] Conjugate[osc1[rtau[tauf-s]]]

here tau_f=-4/3 constant. M=1. The first part is an amplitude, the osc1 satisfies the linear ODE given above. I think this has Levin Potential if I can work out how to input the LevinRules given the above ODE? (Here and in the above I fix my parameters the ODE depends on to (1/10,1) to simplify giving the ICs but I don't that detracts from the main problem)