Integration question taking too long to solve: $\int \frac{(1+x^{4})^{1/2}}{x^{4}}dx$

Question

I want to integrate

$$\int \frac{(1+x^{4})^{1/2}}{x^{4}}dx$$

Please help with this question. It is taking too long to solve. What is the most efficient method?

Answer 1

Integration by parts leads to ... a partial answer !

$$\int\frac{\sqrt{1+x^4}}{x^4}\,dx=-\frac{\sqrt{1+x^4}}{3x^3}+\frac{2}{3}\int\frac{dx}{\sqrt{1+x^4}}$$

But now, this last integral requires elliptic functions ...