Antiderivative of $\frac{e^x}{\sqrt{1-x^2}}$

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Can anyone help me find the following indefinite integral:

$$\int{\frac{e^x}{\sqrt{1-x^2}} dx}$$

I cannot think of any transformation...

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I cannot think of any transformation

No wonder you can't, since even its definite counterpart requires the presence of special Bessel and Struve functions:

$$\int_0^1\frac{e^x}{\sqrt{1-x^2}}~dx~=~\frac\pi2\Big(I_0(1)+L_0(1)\Big),$$

$$\int_{-1}^0\frac{e^x}{\sqrt{1-x^2}}~dx~=~\frac\pi2\Big(I_0(1)-L_0(1)\Big).$$