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$ \int{\frac{\textrm{d}x}{\left( 1+x^2 \right) \sqrt{1+x^4}}} $

I want to use the substitution $x =1/t$,or give another integration $ \int{\frac{\textrm{d}x}{\left( 1-x^2 \right) \sqrt{1+x^4}}} $,but things gets more complex.How to solve this problem?

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    $\begingroup$ The solution will contain an elliptic integral. $\endgroup$ – projectilemotion Mar 16 '17 at 14:35
  • $\begingroup$ Maple says this here $$-{\frac { \left( -1 \right) ^{3/4}\sqrt {1-i{x}^{2}}\sqrt {1+i{x}^{2}} {\it EllipticPi} \left( \sqrt [4]{-1}x,-i,\sqrt {-i}- \left( -1 \right) ^{3/4} \right) }{\sqrt {{x}^{4}+1}}} $$ $\endgroup$ – Dr. Sonnhard Graubner Mar 16 '17 at 14:45
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    $\begingroup$ What is the context? Are you sure you are supposed to be able to find the indefinite integral by hand, or is it perhaps a definite integral? $\endgroup$ – StackTD Mar 16 '17 at 15:15
  • $\begingroup$ Curiously enough, the similar integral $\int{\frac{(1-x^2) \, \textrm{d}x}{\left( 1+x^2 \right) \sqrt{1+x^4}}}$ can be done with elementary functions; see this question. $\endgroup$ – Hans Lundmark Mar 16 '17 at 15:57

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