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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?

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    this is an ellipitic integral. are you familiar with this kind of functions? – 2017-01-19
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    @tired No I am reading this term for the first time.... – 2017-01-19
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    Where can I know more about it ? – 2017-01-19
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    Where did you come up with this integral, if I may ask? – 2017-01-19
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    It was somewhere in my practice book of integration of engineering entrance level , that book doesn't contain the concept of elliptical Integration..But I want to know about it on curiosity basis. – 2017-01-19
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    This seems to be an unlikely problem at that level! Are you sure you copied it correctly? – 2017-01-19
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    *Not to solve it* is very time-saving. Anyway, that is not an elementary integral, it is related with the length of a lemniscate (an elliptic integral). – 2017-01-19
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    @hanslundmark yes correctly , it is not part of my syllabus ,maybe contained in my course by chance. – 2017-01-19

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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 ...