Q: Integral of $\int_0^b\cosh({u^a})du$ Does one calculate the following integral:
$$\int_0^b\cosh({u^a})du$$
where $a \in (0,2]$ and $b>0$.
I found the results from online integral calculator as 
$$=-\frac{\Gamma\left(\frac{1}{a}, \color{red}{-u^a}\right)}{a(-1)^{1/a}} + C.$$
In here, inside the gamma function, as far as I know, there is no any negatif variable. Maybe, I am wrong.
Thanks in advance. 

A: Considering the antiderivative $$I=\int\cosh({u^a})du$$ provided $a>0$, you can also express it in terms of the exponential integral function
$$I=-\frac u{2a} \left(E_{\frac{a-1}{a}}\left(-u^a\right)+E_{\frac{a-1}{a}}\left(u^a\right) \right)$$ which, in turn, can express in terms of the incomplete gamma function.
