The following equation describes thermal cooldown of a body by both convection and radiation:
$\frac{dT}{dt} = a(T-c) + b(T^4-c^4)$
where $T$ is the temperature, $t$ is time and $a$, $b$, $c$ are physical constants. After some conversions and substitutions this leads to the integral
$\int \frac{1}{T^4+eT-f} dT$
where $e$ and $f$ are another representation of the physical constants (after further substitution).
I tried to transform this integral with partial fraction decomposition, but I think that's not possible (at least I failed). I also didn't find any suitable integral in the integral tables out there nor did I find any useful substitutions.
Does anyone know how to solve this integral analytically?
Or: does anybody know that this is not possible?
From a formal point of view. $$\frac {1}{ a(T-c) + b(T^4-c^4)}=\frac 1{(T-c) (a+b (c+T) \left(c^2+T^2\right))}$$ that is to say $$\frac{1}{a+4 b c^3}\left(\frac 1 {T-c} -\frac{3 b c^2+2 b c T+b T^2}{a+b c^3+b c^2 T+b c T^2+b T^3}\right)$$ Now, call $(r_1,r_2,r_3)$ the roots of the cubic polynomial in the second denominator; so $$\frac{3 b c^2+2 b c T+b T^2}{a+b c^3+b c^2 T+b c T^2+b T^3}=\frac{3 c^2+2 c T+T^2}{(T-r_1)(T-r_2)(T-r_3)}$$ that you can decompose using partial fractions and no more problem (in priciple !)