Integrals of the form $e^{-ct^a}(\ln(t))^n dt$

Question

Consider some of the following integrals :

$$\int_1^∞ e^{-t^2} \log^2(t) dt = (1/4)G_{3,4}^{4,0}\left(1\left|\begin{smallmatrix}1,1,1\\ 0,0,0,1/2\end{smallmatrix}\right.\right)$$

$$\int_1^∞ e^{-t^2} \log^3(t) dt = (3/8)G_{4,5}^{5,0}\left(1\left|\begin{smallmatrix}1,1,1,1\\ 0,0,0,0,1/2\end{smallmatrix}\right.\right) $$

$$\int_1^∞ e^{-t^{4/5}} \log^2(t) dt = (125/32)G_{3,4}^{4,0}\left(1\left|\begin{smallmatrix}1,1,1\\ 0,0,0,5/4\end{smallmatrix}\right.\right) $$

$$\int_1^∞ e^{-t^2} \log^5(t) dt = (15/8)G_{6,7}^{7,0}\left(1\left|\begin{smallmatrix}1,1,1,1,1,1\\ 0,0,0,0,0,0,1/2\end{smallmatrix}\right.\right) $$

$$\int_1^∞ e^{-3t^{2/3}} \log^2(t) dt = (3√3/4)G_{3,4}^{4,0}\left(3\left|\begin{smallmatrix}1,1,1\\ 0,0,0,3/2\end{smallmatrix}\right.\right)$$

Here , $G(..)$ is Meijer $G$ function.

(We can see some pattern of the indexes and terms inside the brackets of $G$ function but not of outside)

Also, consider the change of limit

1. $$\int_0^∞ e^{-t^{2/3}}\log^3(t) dt =( -81/64) \sqrt{π} (28 ζ(3) - 6 π^2 + ( \gamma + \log(4)) (2 \gamma ^2 + 3 π^2 + 4 \gamma (\log(4) - 3) + 2 (\log(4) - 6) \log(4)))$$

2. $$\int_0^∞ e^{-t^2} \log^3(t) dt = (-1/32 )\sqrt{π}(28 ζ(3) + 2 \gamma ^3 + 2 \log^3(4) + 3 \gamma (π^2 + 2 \log^2(4)) + 6 \gamma ^2 \log(4) + π^2 \log(64))$$

3. $$\int_0^∞ e^{-t^2}\log^4(t) dt = (1/128 )\sqrt{π} (224 ζ(3) ( \gamma + \log(4)) + 4 \gamma ^4 + 7 π^4 + 4 \log^4(4) + 16 \gamma \log^3(4) + 12 π^2\log^2(4) + 12\gamma ^2 (π^2 + 2 \log^2(4)) + 16\gamma ^3 \log(4) + 8 \gamma π^2\log(64))$$

Generalize the above type of integrals to:

$e^{-ct^a}(\ln(t))^n dt.$

Question: How to evaluate this generalized form for both limits ?

Answer 1

It's not a answer. With Mathematica we have:

For: $n\geq 0$ and $n\in \mathbb{Z}$, $c> 0$, $a>0$

 f[n_, c_, a_] := n!*Exp[-Log[c]/a]*MeijerG[{{},Table[1, n + 1]}, {{Table[0, n + 1], 1/a} // Flatten, {}}, c]/a^(n + 1);
 f[2, 3, 4]
 (*MeijerG[{{}, {1, 1, 1}}, {{0, 0, 0, 1/4}, {}}, 3]/(32 3^(1/4))*)

 g[n_, c_, a_] := Integrate[Exp[-c t^a]*Log[t]^n, {t, 1, Infinity}];
 g[2, 3, 4]
 (*MeijerG[{{}, {1, 1, 1}}, {{0, 0, 0, 1/4}, {}}, 3]/(32 3^(1/4))*)

Answer both are correct.