Integral involving Erf function

Question

I'm struggling with the following integral:

$$ I = \int_{a}^{b} {\frac{\mathrm{Erf}\left(\,{x/c}\,\right)}{\,\sqrt {\,{1 - {x^2}}\,}\,}\,\mathrm{d}x} $$

Honestly, I do not know any approaches to solve it, except trying with Mathematica and searching for a possible solution in tables of integrals involving the Erf function, but all failed.

Can somebody give me a hint?

Thank you very much.

Best regards.

Answer 1

As Robert Israel answered, it does not seem that the antiderivative could be computed even using special functions.

However, considering $$I=\int_0^a\frac{\text{erf}\left(\frac{x}{c}\right)}{\sqrt{1-x^2}}\,dx$$ hoping that $a$ is not too large, you could expand the integrand as a truncated Taylor series and integrate termwise. Otherwise, numerical integration would be required.

You would get something like $$\frac{\text{erf}\left(\frac{x}{c}\right)}{\sqrt{1-x^2}}=\frac{2 x}{\sqrt{\pi } c}+\frac{\left(3 c^2-2\right) x^3}{3 \sqrt{\pi } c^3}+\frac{\left(45 c^4-20 c^2+12\right) x^5}{60 \sqrt{\pi } c^5}+\frac{\left(525 c^6-210 c^4+84 c^2-40\right) x^7}{840 \sqrt{\pi } c^7}+O\left(x^9\right)$$

Let us try using $a=\frac 12$ for various values of $c$ $$\left( \begin{array}{ccc} c & \text{exact} & \text{approximation} \\ 1 & 0.1450370 & 0.14500970 \\ 2 & 0.0747909 & 0.07477398 \\ 3 & 0.0501538 & 0.05014195 \\ 4 & 0.0376931 & 0.03768400 \\ 5 & 0.0301833 & 0.03017601 \\ 6 & 0.0251659 & 0.02515974 \\ 7 & 0.0215775 & 0.02157225 \\ 8 & 0.0188842 & 0.01887956 \\ 9 & 0.0167883 & 0.01678417 \end{array} \right)$$

Another solution would be to use, as Robert Israel answered, integration by parts $$J=\int\frac{\text{erf}\left(\frac{x}{c}\right)}{\sqrt{1-x^2}}\,dx=\sin ^{-1}(x) \text{erf}\left(\frac{x}{c}\right)-\frac{2}{\sqrt{\pi } c}\int { e^{-\frac{x^2}{c^2}} \sin ^{-1}(x)}\,dx$$ $$K=\int { e^{-\frac{x^2}{c^2}} \sin ^{-1}(x)}\,dx=c\int e^{-t^2} \sin ^{-1}(c t)\,dt$$ Using the Taylor expansion $$\sin ^{-1}(c t)=\sum^{\infty}_{n=0} \frac{(2n)!\,c^{2n+1}}{4^n (n!)^2 (2n+1)} t^{2n+1}\qquad \text{for}\qquad |ct|\leq 1$$ and use $$\int t^{2n+1}e^{-t^2}\,dt=-\frac{1}{2} \Gamma \left(n+1,t^2\right)$$

Answer 2

Since $a$ and $b$ are unspecified, you're looking for an antiderivative. Integration by parts gives

$$ \arcsin(x) \text{erf}(x/c) - \frac{2}{c \sqrt{\pi}} \int \arcsin(x)\; e^{-x^2/c^2}\; dx $$

But this does not seem to have an elementary antiderivative.