What is the solution to this heavy integral?

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The integral I=\int \frac{\sqrt{\sqrt{x^4+1}-x^2}}{x^4+1}\,{\rm dx} can be simplified by substituting x^2 = \sinh t, leading to I=\int \frac{\sqrt{\coth t -1}}{\cosh t} dt. Using Mathematica may yield a complex expression involving elliptic integrals, but it has been determined that the integral can actually be expressed in terms of elementary functions. The substitution method proved effective in solving the integral. The discussion highlights the importance of strategic substitutions in integral calculus.
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Try to solve this integral:

I=\int \frac{\sqrt{\sqrt{x^4+1}-x^2}}{x^4+1}\,{\rm dx}
 
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Subbing x^2 =\sinh t will give you something like this

I=\int \frac{\sqrt{\coth t -1}}{\cosh t}{}dt

Feed it now to Mathematica. There's about 50% chances it will return a complicated expression in terms of elliptic integrals.
 
Nice substitution, I managed to solve it now.
Thanks
 
Apparently the integral is expressible in terms of elementary functions.
 

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