What is the convergence radius of the Mac-Laurin-series of
$$f(x)=\frac{x+\sin(x^2)}{e^{x^2}+x^4}\cdot 2x$$ ?
Motivation : I would like to study the number $$\int_0^1 \frac{\sqrt{t}+\sin(t)}{e^t+t^2}dt$$ , which can be transformed to $$\int_0^1 \frac{x+\sin{x^2}}{e^{x^2}+x^4}\cdot 2x\ dx$$ by the substitution $t=x^2$.
I would like to calculate the value of the integral with high precision ($10^5$ digits or more) , so I wonder whether the Mac-Laurin-series of the integrand converges in $[0,1]$. If not, does someone know another method ?
The PARI/GP-algorithm fails already, if $15000$ digits are required (at least if the stack memory is limited to $256MB$).
The radius of convergence is the least absolute value of a root (in the complex plane) of the denominator $e^{x^2} + x^4$. Those roots are $ \pm \sqrt{2 W(\pm i/2)}$, where $W$ is the Lambert W function. The numerical value is approximately $.9219172370$. Since this is less than $1$, the series does not converge on $[0,1]$.