trying to work out some sage math

var('x x0 t k theta r_M d1 d2 d3 d4 a_U r_US v_M f T')
x = 300000*units.length.kilometer
x0 = 299000*units.length.kilometer
v_M = 200*units.length.kilometer/units.time.second^2
r_M = 1*units.length.meter
r_US = 1
theta = 5*10^-9*units.angles.radian
f = 1250
r_a(x) = x * tan(theta) + r_M
r_U(x) = r_US + 0.5 * a_U * (d1 + d2 + d3 + d4)^2
a_U = 50*units.length.meter/units.time.second^2
DAM1(x) = k/x^2 # laser wider than the missile
DAM(x) = DAM1(x) * r_a(x)^2 / r_U(x)^2
TDAM_indefinite(t) = integral(f * DAM(x0 - v_M * t), t)
TDAM(T) = TDAM_indefinite(T) - TDAM_indefinite(0) # between time 0 and time T, how much damage has been done?
TDAM(T)
solve(TDAM(T) - TDAM(0) == 10, T) # solve for the time when 10 damage has been done. Very nasty looking formula.