Optimal. Leaf size=72 \[ \frac {\sqrt {x^4+1}}{2 \left (x^2+x+1\right )}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+1}+x^2-x+1}\right )-\frac {3}{2} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+1}+x^2+x+1}\right ) \]
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Rubi [C] time = 9.86, antiderivative size = 2595, normalized size of antiderivative = 36.04, number of steps used = 248, number of rules used = 20, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.606, Rules used = {6728, 1729, 1209, 1198, 220, 1196, 1217, 1707, 1248, 735, 844, 215, 725, 206, 6742, 2153, 1227, 733, 204, 1336}
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Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 204
Rule 206
Rule 215
Rule 220
Rule 725
Rule 733
Rule 735
Rule 844
Rule 1196
Rule 1198
Rule 1209
Rule 1217
Rule 1227
Rule 1248
Rule 1336
Rule 1707
Rule 1729
Rule 2153
Rule 6728
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{\left (1-x+x^2\right ) \left (1+x+x^2\right )^2} \, dx &=\int \left (\frac {(1+x) \sqrt {1+x^4}}{4 \left (1-x+x^2\right )}+\frac {(-1-2 x) \sqrt {1+x^4}}{2 \left (1+x+x^2\right )^2}+\frac {(-3-x) \sqrt {1+x^4}}{4 \left (1+x+x^2\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {(1+x) \sqrt {1+x^4}}{1-x+x^2} \, dx+\frac {1}{4} \int \frac {(-3-x) \sqrt {1+x^4}}{1+x+x^2} \, dx+\frac {1}{2} \int \frac {(-1-2 x) \sqrt {1+x^4}}{\left (1+x+x^2\right )^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {1+x^4}}{-1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {1+x^4}}{-1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{4} \int \left (\frac {\left (-1+\frac {5 i}{\sqrt {3}}\right ) \sqrt {1+x^4}}{1-i \sqrt {3}+2 x}+\frac {\left (-1-\frac {5 i}{\sqrt {3}}\right ) \sqrt {1+x^4}}{1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{2} \int \left (-\frac {\sqrt {1+x^4}}{\left (1+x+x^2\right )^2}-\frac {2 x \sqrt {1+x^4}}{\left (1+x+x^2\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt {1+x^4}}{\left (1+x+x^2\right )^2} \, dx\right )+\frac {1}{4} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{-1-i \sqrt {3}+2 x} \, dx+\frac {1}{4} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{-1+i \sqrt {3}+2 x} \, dx+\frac {1}{12} \left (-3+5 i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1-i \sqrt {3}+2 x} \, dx-\frac {1}{12} \left (3+5 i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{1+i \sqrt {3}+2 x} \, dx-\int \frac {x \sqrt {1+x^4}}{\left (1+x+x^2\right )^2} \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {4 \sqrt {1+x^4}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}+\frac {4 i \sqrt {1+x^4}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {4 \sqrt {1+x^4}}{3 \left (1+i \sqrt {3}+2 x\right )^2}+\frac {4 i \sqrt {1+x^4}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx\right )+\frac {1}{2} \left (-1-i \sqrt {3}\right ) \int \frac {x \sqrt {1+x^4}}{\left (-1+i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{2} \left (-1+i \sqrt {3}\right ) \int \frac {x \sqrt {1+x^4}}{\left (-1-i \sqrt {3}\right )^2-4 x^2} \, dx-\frac {1}{3} \left (-3+2 i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{\left (1+i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{3} \left (3+2 i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{\left (1-i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{6} \left (3-5 i \sqrt {3}\right ) \int \frac {x \sqrt {1+x^4}}{\left (1-i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{6} \left (3+5 i \sqrt {3}\right ) \int \frac {x \sqrt {1+x^4}}{\left (1+i \sqrt {3}\right )^2-4 x^2} \, dx-\int \frac {\sqrt {1+x^4}}{\left (-1-i \sqrt {3}\right )^2-4 x^2} \, dx-\int \frac {\sqrt {1+x^4}}{\left (-1+i \sqrt {3}\right )^2-4 x^2} \, dx-\int \left (-\frac {2 \left (-1+i \sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}-\frac {2 i \sqrt {1+x^4}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {2 \left (-1-i \sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (1+i \sqrt {3}+2 x\right )^2}-\frac {2 i \sqrt {1+x^4}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx\\ &=\frac {1}{16} \int \frac {\left (-1-i \sqrt {3}\right )^2+4 x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \int \frac {\left (-1+i \sqrt {3}\right )^2+4 x^2}{\sqrt {1+x^4}} \, dx+\frac {2}{3} \int \frac {\sqrt {1+x^4}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {2}{3} \int \frac {\sqrt {1+x^4}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx+\frac {1}{4} \left (-1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (-1+i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )-\frac {1}{2} \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+x^4}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {1}{4} \left (-1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (-1-i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {1+x^4}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {1}{48} \left (3-2 i \sqrt {3}\right ) \int \frac {\left (1+i \sqrt {3}\right )^2+4 x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{48} \left (3+2 i \sqrt {3}\right ) \int \frac {\left (1-i \sqrt {3}\right )^2+4 x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{12} \left (3-5 i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (1-i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )+\frac {1}{6} \left (-3+5 i \sqrt {3}\right ) \int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{12} \left (3+5 i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (1+i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )-\frac {1}{6} \left (3+5 i \sqrt {3}\right ) \int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}-2 \left (\frac {1}{4} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )+\frac {2}{3} \int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (-i+\sqrt {3}-2 i x^2\right )^2}+\frac {\left (1-i \sqrt {3}\right ) x \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2}-\frac {x^2 \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2}\right ) \, dx+\frac {2}{3} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (i+\sqrt {3}+2 i x^2\right )^2}+\frac {\left (1+i \sqrt {3}\right ) x \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2}-\frac {x^2 \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2}\right ) \, dx-\frac {\left (3 i-5 \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{12 \left (i-\sqrt {3}\right )}-\frac {\left (3 i-5 \sqrt {3}\right ) \int \frac {1+x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i-\sqrt {3}\right )}+\frac {1}{16} \left (1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {-4+2 \left (1-i \sqrt {3}\right ) x}{\left (\left (-1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{8} \left (1-i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (-i+\sqrt {3}-2 i x^2\right )^2}+\frac {\left (1-i \sqrt {3}\right ) x \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2}-\frac {x^2 \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2}\right ) \, dx-\frac {1}{24} \left (9-i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \left (1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {-4+2 \left (1+i \sqrt {3}\right ) x}{\left (\left (-1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{8} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (i+\sqrt {3}+2 i x^2\right )^2}+\frac {\left (1+i \sqrt {3}\right ) x \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2}-\frac {x^2 \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2}\right ) \, dx-\frac {1}{24} \left (9+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{12} \left (-3+2 i \sqrt {3}\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{12} \left (3+2 i \sqrt {3}\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{48} \left (-3+5 i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {-4+2 \left (1+i \sqrt {3}\right ) x}{\left (\left (1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {-4-\left (1+i \sqrt {3}\right )^2 x}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{4 \left (i+\sqrt {3}\right )}-\frac {\left (i-\sqrt {3}\right ) \int \frac {1+x^2}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx}{i+\sqrt {3}}-\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{4 \left (i-\sqrt {3}\right )}-\frac {\left (i+\sqrt {3}\right ) \int \frac {1+x^2}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx}{i-\sqrt {3}}-\frac {\left (3 i+5 \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{12 \left (i+\sqrt {3}\right )}-\frac {\left (3 i+5 \sqrt {3}\right ) \int \frac {1+x^2}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i+\sqrt {3}\right )}\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {\left (3-2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (3+2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (3-5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{24 \left (1+i \sqrt {3}\right )}+\frac {\left (3+5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )^2}+\frac {\left (3-2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (3+2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-2 \left (-\frac {x \sqrt {1+x^4}}{4 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\right )-\frac {\left (3 i-5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (9-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (9+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {2}{3} \int \frac {x^2 \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx-\frac {2}{3} \int \frac {x^2 \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx-\frac {4}{3} \int \frac {\sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx-\frac {4}{3} \int \frac {\sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {1}{16} \left (1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{4} \left (1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {x \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {x^2 \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx+\frac {1}{48} \left (9-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{12} \left (9-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{16} \left (1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{4} \left (1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {x \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {x^2 \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {x \sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx-\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )^2\right ) \int \frac {x \sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {1}{48} \left (9+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{12} \left (9+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {i x \sqrt {1+x^4}}{6 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {2 x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}\right ) \left (i-\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{6 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {2 x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}\right ) \left (i+\sqrt {3}+2 i x^2\right )}-\frac {\left (3-2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (3+2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {1}{16} \left (1-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (3-5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{24 \left (1+i \sqrt {3}\right )}+\frac {\left (3+5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )^2}+\frac {\left (3-2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (3+2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-2 \left (-\frac {x \sqrt {1+x^4}}{4 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\right )-\frac {\left (3 i-5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (9-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (9+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {1}{24} i \int \frac {i+\sqrt {3}-2 i x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{24} i \int \frac {-i+\sqrt {3}+2 i x^2}{\sqrt {1+x^4}} \, dx-\frac {2}{3} \int \left (\frac {i \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (-i+\sqrt {3}-2 i x^2\right )^2}+\frac {i \sqrt {1+x^4}}{2 \left (-i+\sqrt {3}-2 i x^2\right )}\right ) \, dx-\frac {2}{3} \int \left (-\frac {i \left (-i-\sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (i+\sqrt {3}+2 i x^2\right )^2}-\frac {i \sqrt {1+x^4}}{2 \left (i+\sqrt {3}+2 i x^2\right )}\right ) \, dx-\frac {\int \frac {-i+\sqrt {3}+2 i x^2}{\sqrt {1+x^4}} \, dx}{6 \left (i-\sqrt {3}\right )}+\frac {1}{4} \left (-1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{16+\left (-1-i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-4-\left (-1-i \sqrt {3}\right )^2 x^2}{\sqrt {1+x^4}}\right )+\frac {1}{3} \left (1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (-i+\sqrt {3}-2 i x\right )^2} \, dx,x,x^2\right )+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \left (\frac {i \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (-i+\sqrt {3}-2 i x^2\right )^2}+\frac {i \sqrt {1+x^4}}{2 \left (-i+\sqrt {3}-2 i x^2\right )}\right ) \, dx+\frac {1}{12} \left (-9+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{16+\left (1-i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-4-\left (1-i \sqrt {3}\right )^2 x^2}{\sqrt {1+x^4}}\right )+\frac {1}{4} \left (-1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{16+\left (-1+i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-4-\left (-1+i \sqrt {3}\right )^2 x^2}{\sqrt {1+x^4}}\right )+\frac {1}{3} \left (1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (i+\sqrt {3}+2 i x\right )^2} \, dx,x,x^2\right )+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \left (-\frac {i \left (-i-\sqrt {3}\right ) \sqrt {1+x^4}}{2 \left (i+\sqrt {3}+2 i x^2\right )^2}-\frac {i \sqrt {1+x^4}}{2 \left (i+\sqrt {3}+2 i x^2\right )}\right ) \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (-i+\sqrt {3}-2 i x\right )^2} \, dx,x,x^2\right )-\frac {1}{3} \left (1+i \sqrt {3}\right )^2 \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\left (i+\sqrt {3}+2 i x\right )^2} \, dx,x,x^2\right )-\frac {1}{12} \left (9+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{16+\left (1+i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-4-\left (1+i \sqrt {3}\right )^2 x^2}{\sqrt {1+x^4}}\right )+\frac {\int \frac {i+\sqrt {3}-2 i x^2}{\sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}+\frac {1}{12} \left (-3 i+\sqrt {3}\right ) \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{12} \left (3 i+\sqrt {3}\right ) \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx-\frac {\left (4+\left (-i+\sqrt {3}\right )^2\right ) \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (-i+\sqrt {3}\right )}-\frac {\left (4+\left (i+\sqrt {3}\right )^2\right ) \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{6 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {2 x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}\right ) \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i+\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{6 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {2 x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}\right ) \left (i+\sqrt {3}+2 i x^2\right )}-\frac {\left (3-2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (3+2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {1}{16} \left (1-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (3-5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{24 \left (1+i \sqrt {3}\right )}+\frac {\left (3+5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )^2}+\frac {\left (1+i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1-i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1-i \sqrt {3}\right )}}+\frac {\left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1+i \sqrt {3}\right )}}-\frac {\left (i-3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1+i \sqrt {3}\right )}}+\frac {\left (i+3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (3-2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (3+2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-2 \left (-\frac {x \sqrt {1+x^4}}{4 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\right )-\frac {\left (3 i-5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (9-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (9+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {1}{3} i \int \frac {\sqrt {1+x^4}}{-i+\sqrt {3}-2 i x^2} \, dx+\frac {1}{3} i \int \frac {\sqrt {1+x^4}}{i+\sqrt {3}+2 i x^2} \, dx-2 \left (\frac {1}{12} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )-\frac {4}{3} \int \frac {\sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx-\frac {4}{3} \int \frac {\sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {1}{3} \left (-i-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (i+\sqrt {3}+2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (i-\sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{i+\sqrt {3}+2 i x^2} \, dx+\frac {1}{3} \left (i-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-i+\sqrt {3}-2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{4 \left (3 i-\sqrt {3}\right )}-\frac {\left (3 i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i-\sqrt {3}\right )}-\frac {1}{3} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{\left (-i+\sqrt {3}-2 i x^2\right )^2} \, dx+\frac {\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx}{3 \left (1-i \sqrt {3}\right )}-\frac {1}{24} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{3} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{\left (i+\sqrt {3}+2 i x^2\right )^2} \, dx+\frac {\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx}{3 \left (1+i \sqrt {3}\right )}+\frac {1}{24} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {\left (1+i \sqrt {3}\right ) \int \frac {1+x^2}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{2 \left (3 i-\sqrt {3}\right )}+\frac {\left (3+i \sqrt {3}\right ) \int \frac {1+x^2}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i-\sqrt {3}\right )}+\frac {1}{6} \left (-i+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (i+\sqrt {3}+2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}-\frac {\left (3-i \sqrt {3}\right ) \int \frac {1+x^2}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i+\sqrt {3}\right )}+\frac {1}{6} \left (i+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-i+\sqrt {3}-2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left (i+\sqrt {3}\right ) \int \frac {\sqrt {1+x^4}}{-i+\sqrt {3}-2 i x^2} \, dx-\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i-\sqrt {3}\right )}+\frac {\left (1-i \sqrt {3}\right ) \int \frac {1+x^2}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{2 \left (3 i+\sqrt {3}\right )}+\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{4 \left (3 i+\sqrt {3}\right )}-\frac {\left (3 i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {4 x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}\right ) \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i+\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {4 x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}\right ) \left (i+\sqrt {3}+2 i x^2\right )}-\frac {x \sqrt {1+x^4}}{3 \left (1-i \sqrt {3}\right ) \left (1+x^2\right )}-\frac {x \sqrt {1+x^4}}{3 \left (1+i \sqrt {3}\right ) \left (1+x^2\right )}-\frac {\left (3-2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (3+2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {1}{16} \left (1-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (3 i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )}+\frac {\left (3-5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{24 \left (1+i \sqrt {3}\right )}+\frac {\left (3+5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )^2}-\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{8 \left (3 i-\sqrt {3}\right )}-\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{8 \left (3 i+\sqrt {3}\right )}-\frac {\left (3 i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i+\sqrt {3}\right )}+\frac {\left (1+i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1-i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1-i \sqrt {3}\right )}}+\frac {\left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1+i \sqrt {3}\right )}}-\frac {\left (i-3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1+i \sqrt {3}\right )}}+\frac {\left (i+3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (3-2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (3+2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-2 \left (-\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}\right )-2 \left (-\frac {x \sqrt {1+x^4}}{4 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\right )-\frac {\left (3 i-5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (9-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (9+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {5 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {5 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {1+x^4}}+\frac {\left (5 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {1}{24} i \int \frac {i+\sqrt {3}-2 i x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{24} i \int \frac {-i+\sqrt {3}+2 i x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{12} i \int \frac {i+\sqrt {3}-2 i x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{12} i \int \frac {-i+\sqrt {3}+2 i x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{3} i \operatorname {Subst}\left (\int \frac {1}{\left (-i+\sqrt {3}-2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{3} i \operatorname {Subst}\left (\int \frac {1}{\left (i+\sqrt {3}+2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {2}{3} i \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx-\frac {2}{3} i \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx-\frac {\int \frac {-i+\sqrt {3}+2 i x^2}{\sqrt {1+x^4}} \, dx}{6 \left (i-\sqrt {3}\right )}-\frac {1}{12} \left (i-\sqrt {3}\right ) \int \frac {i+\sqrt {3}-2 i x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{6} \left (i-\sqrt {3}\right ) \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{12} \left (3 i-\sqrt {3}\right ) \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{12} \left (-1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{6} \left (-1-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{12} \left (-1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{6} \left (-1+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{3} \left (-i+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i+\sqrt {3}+2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {\int \frac {i+\sqrt {3}-2 i x^2}{\sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}+\frac {1}{12} \left (i+\sqrt {3}\right ) \int \frac {-i+\sqrt {3}+2 i x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{6} \left (i+\sqrt {3}\right ) \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{3} \left (i+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+\sqrt {3}-2 i x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )+\frac {1}{12} \left (3 i+\sqrt {3}\right ) \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx-\frac {\left (4+\left (-i+\sqrt {3}\right )^2\right ) \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (-i+\sqrt {3}\right )}-\frac {\left (4+\left (i+\sqrt {3}\right )^2\right ) \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {4 x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}\right ) \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i+\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {4 x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}\right ) \left (i+\sqrt {3}+2 i x^2\right )}-\frac {x \sqrt {1+x^4}}{3 \left (1-i \sqrt {3}\right ) \left (1+x^2\right )}-\frac {x \sqrt {1+x^4}}{3 \left (1+i \sqrt {3}\right ) \left (1+x^2\right )}-\frac {\left (3-2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (3+2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {3}{16} \left (1-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )-\frac {3}{16} \left (1+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (3 i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )}+\frac {\left (3-5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{24 \left (1+i \sqrt {3}\right )}+\frac {\left (3+5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )^2}-\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{8 \left (3 i-\sqrt {3}\right )}-\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{8 \left (3 i+\sqrt {3}\right )}-\frac {\left (3 i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i+\sqrt {3}\right )}+\frac {\left (1+i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1-i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1-i \sqrt {3}\right )}}+\frac {\left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1+i \sqrt {3}\right )}}-\frac {\left (i-3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1+i \sqrt {3}\right )}}+\frac {\left (i+3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (3-2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (3+2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-2 \left (-\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}\right )-2 \left (-\frac {x \sqrt {1+x^4}}{4 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\right )-\frac {\left (3 i-5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (9-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-\frac {\left (9+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {5 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {5 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {1+x^4}}+\frac {\left (5 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {1}{3} i \operatorname {Subst}\left (\int \frac {1}{-4+\left (-i+\sqrt {3}\right )^2-x^2} \, dx,x,\frac {-2 i-\left (-i+\sqrt {3}\right ) x^2}{\sqrt {1+x^4}}\right )-\frac {1}{3} i \operatorname {Subst}\left (\int \frac {1}{-4+\left (i+\sqrt {3}\right )^2-x^2} \, dx,x,\frac {2 i-\left (i+\sqrt {3}\right ) x^2}{\sqrt {1+x^4}}\right )-2 \left (\frac {1}{12} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {1}{6} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )+\frac {1}{3} \left (-i-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{-4+\left (-i+\sqrt {3}\right )^2-x^2} \, dx,x,\frac {-2 i-\left (-i+\sqrt {3}\right ) x^2}{\sqrt {1+x^4}}\right )+\frac {1}{3} \left (i-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{-4+\left (i+\sqrt {3}\right )^2-x^2} \, dx,x,\frac {2 i-\left (i+\sqrt {3}\right ) x^2}{\sqrt {1+x^4}}\right )-\frac {(4 i) \int \frac {1+x^2}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3 i-\sqrt {3}\right )}+\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{4 \left (3 i-\sqrt {3}\right )}-\frac {\left (3 i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i-\sqrt {3}\right )}-\frac {1}{6} \left (-1-i \sqrt {3}\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx}{3 \left (1-i \sqrt {3}\right )}+\frac {1}{6} \left (1-i \sqrt {3}\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {(2 i) \int \frac {1}{\sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3-i \sqrt {3}\right )}-\frac {1}{24} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{12} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{6} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx}{3 \left (1+i \sqrt {3}\right )}+\frac {1}{24} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{12} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{6} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {\left (1+i \sqrt {3}\right ) \int \frac {1+x^2}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{2 \left (3 i-\sqrt {3}\right )}+\frac {(2 i) \int \frac {1}{\sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3+i \sqrt {3}\right )}+\frac {\left (3+i \sqrt {3}\right ) \int \frac {1+x^2}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i-\sqrt {3}\right )}-\frac {\left (-i+\sqrt {3}\right ) \int \frac {1+x^2}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3 i-\sqrt {3}\right )}-\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}-\frac {\left (3-i \sqrt {3}\right ) \int \frac {1+x^2}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i+\sqrt {3}\right )}-\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i-\sqrt {3}\right )}-\frac {(4 i) \int \frac {1+x^2}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt {3} \left (3 i+\sqrt {3}\right )}+\frac {\left (3 i-\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (3 i+\sqrt {3}\right )}+\frac {\left (1-i \sqrt {3}\right ) \int \frac {1+x^2}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{2 \left (3 i+\sqrt {3}\right )}+\frac {\left (3+i \sqrt {3}\right ) \int \frac {1+x^2}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (3 i+\sqrt {3}\right )}+\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{4 \left (3 i+\sqrt {3}\right )}+\frac {\left (3 i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (3 i-\sqrt {3}\right )}-\frac {\left (3 i+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{6 \left (i+\sqrt {3}\right )}\\ &=\frac {1}{16} \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {1}{16} \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3-5 i \sqrt {3}\right ) \sqrt {1+x^4}-\frac {1}{48} \left (3+5 i \sqrt {3}\right ) \sqrt {1+x^4}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}+2 i x^2\right )}-\frac {4 x \sqrt {1+x^4}}{3 \left (i-\sqrt {3}\right ) \left (i-\sqrt {3}+2 i x^2\right )}+\frac {\left (i-\sqrt {3}\right ) \sqrt {1+x^4}}{6 \left (i+\sqrt {3}+2 i x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {i x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}+2 i x^2\right )}-\frac {4 x \sqrt {1+x^4}}{3 \left (i+\sqrt {3}\right ) \left (i+\sqrt {3}+2 i x^2\right )}-\frac {2 x \sqrt {1+x^4}}{3 \left (1-i \sqrt {3}\right ) \left (1+x^2\right )}-\frac {\left (1-i \sqrt {3}\right ) x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {2 x \sqrt {1+x^4}}{3 \left (1+i \sqrt {3}\right ) \left (1+x^2\right )}-\frac {\left (1+i \sqrt {3}\right ) x \sqrt {1+x^4}}{6 \left (1+x^2\right )}-\frac {\left (3-2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {\left (3+2 i \sqrt {3}\right ) x \sqrt {1+x^4}}{12 \left (1+x^2\right )}-\frac {3}{16} \left (1-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9-i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )-\frac {3}{16} \left (1+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {1}{48} \left (9+i \sqrt {3}\right ) \sinh ^{-1}\left (x^2\right )+\frac {5}{12} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\left (3 i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{6 \left (i-\sqrt {3}\right )}-\frac {1}{12} \left (1-i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {1}{12} \left (1+i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )+\frac {\left (3-5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{24 \left (1+i \sqrt {3}\right )}+\frac {\left (3+5 i \sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{12 \left (i-\sqrt {3}\right )^2}-\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{4 \left (3 i-\sqrt {3}\right )}-\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{4 \left (3 i+\sqrt {3}\right )}-\frac {\left (3 i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )}{6 \left (i+\sqrt {3}\right )}+\frac {i \tan ^{-1}\left (\frac {2 i-\left (i-\sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{3 \sqrt {2 \left (1+i \sqrt {3}\right )}}-\frac {\left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {2 i-\left (i-\sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{3 \sqrt {2 \left (1+i \sqrt {3}\right )}}+\frac {i \tan ^{-1}\left (\frac {2 i-\left (i+\sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{3 \sqrt {2 \left (1-i \sqrt {3}\right )}}-\frac {\left (i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {2 i-\left (i+\sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{3 \sqrt {2 \left (1-i \sqrt {3}\right )}}+\frac {\left (1+i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1-i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1-i \sqrt {3}\right )}}+\frac {\left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {2 \left (1+i \sqrt {3}\right )}}-\frac {\left (i-3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1+i \sqrt {3}\right )}}+\frac {\left (i+3 \sqrt {3}\right ) \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{8 \sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (1-i \sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (1+i \sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}+\frac {\left (3-2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}+\frac {\left (3+2 i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}-4 \left (-\frac {x \sqrt {1+x^4}}{12 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}\right )-2 \left (-\frac {x \sqrt {1+x^4}}{6 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}\right )-2 \left (-\frac {x \sqrt {1+x^4}}{4 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\right )-\frac {\left (3 i-5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {3} \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {11 \left (1-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {\left (9-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}+\frac {11 \left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {\left (9+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{48 \sqrt {1+x^4}}-\frac {7 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {7 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {3} \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (3 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (3 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3 i+5 \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}-\frac {i \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {3} \sqrt {1+x^4}}+\frac {\left (5 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (3+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{24 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {3} \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \left (i-\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^4}}+\frac {\left (5 i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}\\ \end {align*}
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Mathematica [C] time = 1.44, size = 350, normalized size = 4.86 \begin {gather*} \frac {1}{8} \left (\frac {4 \sqrt {x^4+1}}{x^2+x+1}-i \sqrt {6+6 i \sqrt {3}} \tanh ^{-1}\left (\frac {2+\left (-1-i \sqrt {3}\right ) x^2}{\sqrt {2+2 i \sqrt {3}} \sqrt {x^4+1}}\right )-\sqrt {2+2 i \sqrt {3}} \tanh ^{-1}\left (\frac {2+\left (-1-i \sqrt {3}\right ) x^2}{\sqrt {2+2 i \sqrt {3}} \sqrt {x^4+1}}\right )+i \sqrt {6-6 i \sqrt {3}} \tanh ^{-1}\left (\frac {2+i \left (\sqrt {3}+i\right ) x^2}{\sqrt {2-2 i \sqrt {3}} \sqrt {x^4+1}}\right )-\sqrt {2-2 i \sqrt {3}} \tanh ^{-1}\left (\frac {2+i \left (\sqrt {3}+i\right ) x^2}{\sqrt {2-2 i \sqrt {3}} \sqrt {x^4+1}}\right )+4 \sqrt [4]{-1} \Pi \left (-\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+4 \sqrt [4]{-1} \Pi \left (-(-1)^{5/6};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right )-\frac {1}{2} \sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.81, size = 72, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^4}}{2 \left (1+x+x^2\right )}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{1-x+x^2+\sqrt {1+x^4}}\right )-\frac {3}{2} \tan ^{-1}\left (\frac {x}{1+x+x^2+\sqrt {1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 73, normalized size = 1.01 \begin {gather*} \frac {3 \, {\left (x^{2} + x + 1\right )} \arctan \left (\frac {\sqrt {x^{4} + 1}}{x^{2} + 2 \, x + 1}\right ) + {\left (x^{2} + x + 1\right )} \arctan \left (\frac {\sqrt {x^{4} + 1}}{x^{2} - 2 \, x + 1}\right ) + 2 \, \sqrt {x^{4} + 1}}{4 \, {\left (x^{2} + x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + x + 1\right )}^{2} {\left (x^{2} - x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.05, size = 198, normalized size = 2.75
method | result | size |
trager | \(\frac {\sqrt {x^{4}+1}}{2 x^{2}+2 x +2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{7}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}-2 \sqrt {x^{4}+1}\, x^{5}+5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-2 x^{4} \sqrt {x^{4}+1}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \sqrt {x^{4}+1}\, x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 x^{2} \sqrt {x^{4}+1}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \sqrt {x^{4}+1}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (x^{2}-x +1\right ) \left (x^{2}+x +1\right )^{3}}\right )}{4}\) | \(198\) |
risch | \(\frac {\sqrt {x^{4}+1}}{2 x^{2}+2 x +2}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {3 \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}+\frac {3 \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}\) | \(539\) |
elliptic | \(-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (\sqrt {3}\, \arctanh \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \sqrt {3}}{2}\right )-9 \arctan \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{\left (\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \left (-x^{2}+1\right )}\right )\right )}{12 \sqrt {\frac {\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1}{\left (\frac {x^{2}+1}{-x^{2}+1}+1\right )^{2}}}\, \left (\frac {x^{2}+1}{-x^{2}+1}+1\right )}+\frac {\left (\frac {3 \sqrt {3}\, \arctanh \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \sqrt {3}}{2}\right ) \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}-\frac {9 \arctan \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{\left (\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \left (-x^{2}+1\right )}\right ) \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+\sqrt {3}\, \arctanh \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \sqrt {3}}{2}\right )+\frac {6 \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{-x^{2}+1}-3 \arctan \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{\left (\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \left (-x^{2}+1\right )}\right )\right ) \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \sqrt {2}}{12 \left (\frac {3 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \sqrt {\frac {\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1}{\left (\frac {x^{2}+1}{-x^{2}+1}+1\right )^{2}}}\, \left (\frac {x^{2}+1}{-x^{2}+1}+1\right )}+\frac {\left (-\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{4 x \left (\frac {x^{4}+1}{2 x^{2}}+\frac {1}{2}\right )}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+1}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(602\) |
default | \(\frac {\sqrt {x^{4}+1}}{2 x^{2}+2 x +2}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {i \sqrt {3}\, \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{3}+\frac {i \sqrt {3}\, \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{3}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}-\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{6}\right ) \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}-\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{6}\right ) \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{4}\) | \(753\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + x + 1\right )}^{2} {\left (x^{2} - x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2-1\right )\,\sqrt {x^4+1}}{\left (x^2-x+1\right )\,{\left (x^2+x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} + 1}}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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