# How to compute the integration $\int{\frac{\textrm{d}x}{\left( 1+x^2 \right) \sqrt{1+x^4}}}$

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

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?

• The solution will contain an elliptic integral. – projectilemotion Mar 16 '17 at 14:35
• Maple says this here $$-{\frac { \left( -1 \right) ^{3/4}\sqrt {1-i{x}^{2}}\sqrt {1+i{x}^{2}} {\it EllipticPi} \left( \sqrt [4]{-1}x,-i,\sqrt {-i}- \left( -1 \right) ^{3/4} \right) }{\sqrt {{x}^{4}+1}}}$$ – Dr. Sonnhard Graubner Mar 16 '17 at 14:45
• 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? – StackTD Mar 16 '17 at 15:15
• 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. – Hans Lundmark Mar 16 '17 at 15:57