How to find $\int e^{2\cos 2x}dx$? I have  found a function. I'm unable to find its integral. I have searched for it in calculus books but couldn't find answer. I just want a hint.
$$\int e^{2\cos 2x}dx.$$
Also about the function...
$$\int e^{x^2}dx.$$
Or any function with complicated power.
 A: Using $$\displaystyle e^x = 1+\frac{x}{1!}+\frac{x^2}{2!}+...........\infty$$
So $$\displaystyle \int e^{x^2}dx = \int \left[1+\frac{x^2}{1!}+\frac{x^4}{2!}+......\infty\right]dx$$
So we get $$\displaystyle \int e^{x^2}dx = x+\frac{x^3}{1!\times 3}+\frac{x^5}{2!\times 5}+.......\infty+\mathcal{C} = \sum^{\infty}_{k=0}\frac{x^{2k+1}}{k!\times (2k+1)}+\mathcal{C}$$
A: Not every indefinite integral has a closed form.
A: Both indefinite integrals cannot be expressed with a finite number of elementary functions.
$$\int e^{x^2}dx = \frac{1}{2} \sqrt{\pi}\:\text{erfi}(x)+constant$$
erfi(x) is a special function : http://mathworld.wolfram.com/Erfi.html
erfi$(x)$ is related to the Dawson's function Daw$(x)$ : http://mathworld.wolfram.com/DawsonsIntegral.html
Both erfi$(x)$ and Daw$(x)$ are particular hypergeometric functions. 
For general information about the uses of special functions : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales
The case of :
$$\int e^{2\cos 2x}dx$$
is even more complicated. As far as I know, there is no convenient standard special function. Use infinite series for further calculus, and/or numerical methods of computation in practice.
Note that the above comment concerns the indefinite integrals. In some particular case of these integrals with particular bounds (particular finite integrals) closed form might exist. For example :
$$\int_0^{\pi} e^{2\cos 2x}dx=\pi\: I_0(2)$$
where $I_0(x)$ is the Modified Bessel function of the first kind. 
A: For $\int e^{2\cos 2x}~dx$ ,
$\int e^{2\cos 2x}~dx$
$=\int e^{4\cos^2x-2}~dx$
$=\int\sum\limits_{n=0}^\infty\dfrac{(-1)^n4^n\cos^{2n}x}{e^2n!}dx$
$=\int\left(e^{-2}+\sum\limits_{n=1}^\infty\dfrac{(-1)^n4^n\cos^{2n}x}{e^2n!}\right)dx$
For $n$ is any natural number,
$\int\cos^{2n}x~dx=\dfrac{(2n)!x}{4^n(n!)^2}+\sum\limits_{k=1}^n\dfrac{(2n)!((k-1)!)^2\sin x~\cos^{2k-1}x}{4^{n-k+1}(n!)^2(2k-1)!}+C$
This result can be done by successive integration by parts.
$\therefore\int\left(e^{-2}+\sum\limits_{n=1}^\infty\dfrac{(-1)^n4^n\cos^{2n}x}{e^2n!}\right)dx$
$=e^{-2}x+\sum\limits_{n=1}^\infty\dfrac{(-1)^n(2n)!x}{e^2(n!)^3}+\sum\limits_{n=1}^\infty\sum\limits_{k=1}^n\dfrac{(-1)^n4^{k-1}(2n)!((k-1)!)^2\sin x~\cos^{2k-1}x}{e^2(n!)^3(2k-1)!}+C$
$=\sum\limits_{n=0}^\infty\dfrac{(-1)^n(2n)!x}{e^2(n!)^3}+\sum\limits_{n=1}^\infty\sum\limits_{k=1}^n\dfrac{(-1)^n4^{k-1}(2n)!((k-1)!)^2\sin x~\cos^{2k-1}x}{e^2(n!)^3(2k-1)!}+C$
