$$\int\sum_{n=0}^\infty\frac1{n!}\left[\int\cos^3t\left(\tan^{n+4}(t)+4\tan^n(t)\right)\,\mathrm dt\right]$$
This is the integral that I want to calculate any ways that you can suggest so as it becomes easier to solve If you are wondering the source of the question then it's actually part of a bigger problem, which was the integration of $$\int \frac{e^x (x^4 + 4)}{(x^2+1)^{5/2}} \,\mathrm{d}x$$ StackExchange's Solution to the First Question