Problem :
Show that :
$$\int_{0}^{1}\Gamma\left(\cot\left(e^{-x^{2}}\right)+1\right)dx<\int_{0}^{1}x^{-x}dx$$
I use Desmos notation .
In hoping to show it by hand I have tried the Sophomore dream :
$$\int_{0}^{1}x^{-x}dx=\sum_{n=1}^{\infty}n^{-n}$$
I cannot find a power series for the other side .
For information we need to show :
$$1.29125<1.29128$$
Wich looks very hard at first glance but a also a very challenging problem .
Some remarks :
Let $0<x<1$ then it seems the $n$ th derivative is strictly positive or :
$$f^{(n)}(x)>0$$
Where :
$$f(x)=\Gamma\left(\cot\left(e^{-x^{2}}\right)+1\right)$$
Wich says (and again it seems) if the power series exists then the coefficients are strictly positives .Moreover around zero it seems there is only the even derivatives wich seems to increase exponentialy .
As point out by ClaudeLeibovici the conjecture is false but it seems true as power series around $x=1$
Any ideas to show it without any calculator so by hand ?
If we expand $f(x)$ as a series$$f(x)=\Gamma\Big[\cot\left(e^{-x^{2}}\right)+1\Big]=\sum_{n=0}^\infty a_n\,x^{2n}$$ the coefficient are not all positive $$\left( \begin{array}{cc} n & a_n \\ 0 & +0.898944 \\ 1 & +0.205125 \\ 2 & +0.796432 \\ 3 & +0.166882 \\ 4 & +0.779911 \\ 5 & -0.144472 & \text{negative}\\ 6 & +0.782758 \\ 7 & -0.480548 & \text{negative}\\ 8 & +0.89023 \\ 9 & -0.833327 & \text{negative}\\ 10 & +1.130030 \\ \end{array} \right)$$