How to integrate $\int \frac{1}{\sqrt{1+29x^2+100x^4}}dx$ and $\int \frac{1}{\sqrt{1-2x^2-8x^4}}dx$ using elliptic functions?

Question

How to integrate $$\int \frac{1}{\sqrt{1+29x^2+100x^4}}dx$$ and $$\int\frac{1}{\sqrt{1-2x^2-8x^4}}dx$$ using elliptic functions?

I have tried to use them, but I got incorrect formula $$\frac{1}{\sqrt{2}}F(arctan(\sqrt{2}x)∣3)$$ for the first one (second argument should be less or equal 1).

Could anyone solve it?

Thanks

Answer 1

I'm not totally sure how to do it by hand, but using the identity

$$\int\frac{1}{\sqrt{a+bx^2+cx^4}}\mathrm{d}x=-\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} F\left(i \sinh ^{-1}\left(\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right)|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right)}{\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}}$$

one obtains

$$\int\frac{1}{\sqrt{1+29x^2+100x^4}}\mathrm{d}x=-\frac{i \sqrt{4 x^2+1} \sqrt{25 x^2+1} F\left(i \sinh ^{-1}(5 x)|\frac{4}{25}\right)}{5 \sqrt{100 x^4+29 x^2+1}}$$

and

$$\int\frac{1}{\sqrt{1-2x^2-8x^4}}\mathrm{d}x=\frac{1}{2} F\left(\sin ^{-1}(2 x)|-\frac{1}{2}\right)$$

which agrees with numerical experiments.