Evaluate in terms of elliptic integral

Question

Define $$f(t):=\int_0^1 \frac{\sqrt{x+t+\sqrt{t^2+2tx+1}}}{\sqrt{1-x^2}} dx $$ Can this integral be evaluated in terms of elliptic integrals? I ask because I have established the functional equation $$f(t)+f(-t)=2\pi \sqrt{2t}$$ And I hope to turn it into a potentially useful functional equation for elliptic integrals.

Thanks!

Answer 1

I am not sure that we could find any closed form for this integral and I suppose that you will need numerical integration to get $$f(t)=\int_0^1 \frac{\sqrt{x+t+\sqrt{t^2+2tx+1}}}{\sqrt{1-x^2}}\, dx$$ On the other hand, I am surprised by $$f(t)+f(-t)=2\pi \sqrt{2t}$$ Consider $t$ to be small; using Taylor expansions for the integrand and integrating, we have $$f(t) =2+t-\frac{t^2}{12}-\frac{t^3}{40}+\frac{5 t^4}{448}+O\left(t^5\right)$$ that is to say $$f(t)+f(-t)\approx 4-\frac{t^2}{6}+\frac{5 t^4}{224}+O\left(t^6\right)$$ which is "almost" exact for $-1 \leq t \leq 1$.

I produce below a table of numerical values $$\left( \begin{array}{cccc} t & f(t) & f(-t) & f(t)+f(-t) & 4-\frac{t^2}{6}+\frac{5 t^4}{224}\\ -3.00 & 0.444240 & 4.25406 & 4.69830 \\ -2.75 & 0.467638 & 4.10797 & 4.57561 \\ -2.50 & 0.495114 & 3.95676 & 4.45187 \\ -2.25 & 0.527996 & 3.79987 & 4.32787 \\ -2.00 & 0.568309 & 3.63669 & 4.20500 \\ -1.75 & 0.619341 & 3.46649 & 4.08583 \\ -1.50 & 0.686924 & 3.28841 & 3.97534 \\ -1.25 & 0.783061 & 3.10149 & 3.88456 \\ -1.00 & 0.945822 & 2.90465 & 3.85047 &3.85565\\ -0.75 & 1.21549 & 2.69676 & 3.91225 &3.91331\\ -0.50 & 1.48276 & 2.47686 & 3.95962 &3.95973\\ -0.25 & 1.74522 & 2.24445 & 3.98967 &3.98967\\ 0.00 & 2.00000 & 2.00000 & 4.00000 &4.00000\\ 0.25 & 2.24445 & 1.74522 & 3.98967 &3.98967\\ 0.50 & 2.47686 & 1.48276 & 3.95962 &3.95973\\ 0.75 & 2.69676 & 1.21549 & 3.91225 &3.91331\\ 1.00 & 2.90465 & 0.945822 & 3.85047& 3.85565\\ 1.25 & 3.10149 & 0.783061 & 3.88456 \\ 1.50 & 3.28841 & 0.686924 & 3.97534 \\ 1.75 & 3.46649 & 0.619341 & 4.08583 \\ 2.00 & 3.63669 & 0.568309 & 4.20500 \\ 2.25 & 3.79987 & 0.527996 & 4.32787 \\ 2.50 & 3.95676 & 0.495114 & 4.45187 \\ 2.75 & 4.10797 & 0.467638 & 4.57561 \\ 3.00 & 4.25406 & 0.44424 & 4.69830 \end{array} \right)$$

If $t$ is large, using Taylor again and integrating

$$f(t)=\frac{\pi \sqrt{t}}{\sqrt{2}}+\frac{1}{\sqrt{2t}}-\frac{1}{24 t^2\sqrt{2t}}+O\left(\frac{1}{t^{9/2}}\right)$$ identical to the expansion of the formula given by Mariusz Iwaniuk in his comment.

For $t=10$, the approximation leads to $7.24833$ while the exact value would be $7.24833$ (!!).