Optimal. Leaf size=53 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {x^4+1}}\right )}{2 \sqrt {3}}-\frac {\sqrt {x^4+1} x}{2 \left (x^4+3 x^2+1\right )} \]
________________________________________________________________________________________
Rubi [C] time = 2.88, antiderivative size = 1165, normalized size of antiderivative = 21.98, number of steps used = 80, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 1227, 1198, 220, 1196, 1217, 1707, 1209, 6728}
result too large to display
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 1196
Rule 1198
Rule 1209
Rule 1217
Rule 1227
Rule 1707
Rule 6728
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx &=\int \left (\frac {\left (-2-3 x^2\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2}+\frac {\sqrt {1+x^4}}{1+3 x^2+x^4}\right ) \, dx\\ &=\int \frac {\left (-2-3 x^2\right ) \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx+\int \frac {\sqrt {1+x^4}}{1+3 x^2+x^4} \, dx\\ &=\int \left (-\frac {2 \sqrt {1+x^4}}{\sqrt {5} \left (-3+\sqrt {5}-2 x^2\right )}-\frac {2 \sqrt {1+x^4}}{\sqrt {5} \left (3+\sqrt {5}+2 x^2\right )}\right ) \, dx+\int \left (-\frac {2 \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2}-\frac {3 x^2 \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx\right )-3 \int \frac {x^2 \sqrt {1+x^4}}{\left (1+3 x^2+x^4\right )^2} \, dx-\frac {2 \int \frac {\sqrt {1+x^4}}{-3+\sqrt {5}-2 x^2} \, dx}{\sqrt {5}}-\frac {2 \int \frac {\sqrt {1+x^4}}{3+\sqrt {5}+2 x^2} \, dx}{\sqrt {5}}\\ &=-\left (2 \int \left (\frac {4 \sqrt {1+x^4}}{5 \left (-3+\sqrt {5}-2 x^2\right )^2}+\frac {4 \sqrt {1+x^4}}{5 \sqrt {5} \left (-3+\sqrt {5}-2 x^2\right )}+\frac {4 \sqrt {1+x^4}}{5 \left (3+\sqrt {5}+2 x^2\right )^2}+\frac {4 \sqrt {1+x^4}}{5 \sqrt {5} \left (3+\sqrt {5}+2 x^2\right )}\right ) \, dx\right )-3 \int \left (\frac {2 \left (-3+\sqrt {5}\right ) \sqrt {1+x^4}}{5 \left (-3+\sqrt {5}-2 x^2\right )^2}-\frac {6 \sqrt {1+x^4}}{5 \sqrt {5} \left (-3+\sqrt {5}-2 x^2\right )}+\frac {2 \left (-3-\sqrt {5}\right ) \sqrt {1+x^4}}{5 \left (3+\sqrt {5}+2 x^2\right )^2}-\frac {6 \sqrt {1+x^4}}{5 \sqrt {5} \left (3+\sqrt {5}+2 x^2\right )}\right ) \, dx+\frac {\int \frac {3+\sqrt {5}-2 x^2}{\sqrt {1+x^4}} \, dx}{2 \sqrt {5}}+\frac {\int \frac {-3+\sqrt {5}+2 x^2}{\sqrt {1+x^4}} \, dx}{2 \sqrt {5}}+\frac {1}{5} \left (3 \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{5} \left (3 \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx\\ &=-\left (\frac {8}{5} \int \frac {\sqrt {1+x^4}}{\left (-3+\sqrt {5}-2 x^2\right )^2} \, dx\right )-\frac {8}{5} \int \frac {\sqrt {1+x^4}}{\left (3+\sqrt {5}+2 x^2\right )^2} \, dx-\frac {8 \int \frac {\sqrt {1+x^4}}{-3+\sqrt {5}-2 x^2} \, dx}{5 \sqrt {5}}-\frac {8 \int \frac {\sqrt {1+x^4}}{3+\sqrt {5}+2 x^2} \, dx}{5 \sqrt {5}}+\frac {18 \int \frac {\sqrt {1+x^4}}{-3+\sqrt {5}-2 x^2} \, dx}{5 \sqrt {5}}+\frac {18 \int \frac {\sqrt {1+x^4}}{3+\sqrt {5}+2 x^2} \, dx}{5 \sqrt {5}}+\frac {1}{5} \left (6 \left (3-\sqrt {5}\right )\right ) \int \frac {\sqrt {1+x^4}}{\left (-3+\sqrt {5}-2 x^2\right )^2} \, dx+\frac {1}{10} \left (5-\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{5} \left (6 \left (3+\sqrt {5}\right )\right ) \int \frac {\sqrt {1+x^4}}{\left (3+\sqrt {5}+2 x^2\right )^2} \, dx+\frac {1}{10} \left (5+\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {\left (3 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{5 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}+\frac {\left (6 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}-\frac {\left (3 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{5 \left (-4+\left (3+\sqrt {5}\right )^2\right )}+\frac {\left (6 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-4+\left (3+\sqrt {5}\right )^2\right )}\\ &=\frac {3 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}+2 x^2\right )}-\frac {4 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}\right ) \left (3-\sqrt {5}+2 x^2\right )}+\frac {3 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}+2 x^2\right )}-\frac {4 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}\right ) \left (3+\sqrt {5}+2 x^2\right )}-\frac {\sqrt {3} \left (1-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{2 \left (5-\sqrt {5}\right )}-\frac {\sqrt {3} \left (1+\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{2 \left (5+\sqrt {5}\right )}-\frac {3 \left (3-\sqrt {5}\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 )}{2 \left (5-\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (5-\sqrt {5}\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 )}{20 \sqrt {1+x^4}}-\frac {3 \left (3+\sqrt {5}\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 )}{2 \left (5+\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (5+\sqrt {5}\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 )}{20 \sqrt {1+x^4}}+\frac {3 \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 {3 \left (5+5 \sqrt {5}\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 )}{20 \left (1+\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {3}{20} \int \frac {3+\sqrt {5}-2 x^2}{\sqrt {1+x^4}} \, dx-\frac {3}{20} \int \frac {-3+\sqrt {5}+2 x^2}{\sqrt {1+x^4}} \, dx+\frac {2 \int \frac {3+\sqrt {5}-2 x^2}{\sqrt {1+x^4}} \, dx}{5 \sqrt {5}}+\frac {2 \int \frac {-3+\sqrt {5}+2 x^2}{\sqrt {1+x^4}} \, dx}{5 \sqrt {5}}-\frac {9 \int \frac {3+\sqrt {5}-2 x^2}{\sqrt {1+x^4}} \, dx}{10 \sqrt {5}}-\frac {9 \int \frac {-3+\sqrt {5}+2 x^2}{\sqrt {1+x^4}} \, dx}{10 \sqrt {5}}+\frac {1}{10} \left (3 \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{25} \left (12 \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{25} \left (27 \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {\int \frac {-3+\sqrt {5}+2 x^2}{\sqrt {1+x^4}} \, dx}{5 \left (-3+\sqrt {5}\right )}-\frac {\int \frac {3+\sqrt {5}-2 x^2}{\sqrt {1+x^4}} \, dx}{5 \left (3+\sqrt {5}\right )}-\frac {1}{10} \left (3 \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{25} \left (12 \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{25} \left (27 \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {\left (-4+\left (-3+\sqrt {5}\right )^2\right ) \int \frac {1}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-3+\sqrt {5}\right )}+\frac {\left (-4+\left (3+\sqrt {5}\right )^2\right ) \int \frac {1}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (3+\sqrt {5}\right )}\\ &=\frac {3 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}+2 x^2\right )}-\frac {4 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}\right ) \left (3-\sqrt {5}+2 x^2\right )}+\frac {3 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}+2 x^2\right )}-\frac {4 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}\right ) \left (3+\sqrt {5}+2 x^2\right )}-\frac {\sqrt {3} \left (1-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{2 \left (5-\sqrt {5}\right )}-\frac {\sqrt {3} \left (1+\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{2 \left (5+\sqrt {5}\right )}-\frac {3 \left (3-\sqrt {5}\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 )}{2 \left (5-\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (5-\sqrt {5}\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 )}{20 \sqrt {1+x^4}}-\frac {3 \left (3+\sqrt {5}\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 )}{2 \left (5+\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (5+\sqrt {5}\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 )}{20 \sqrt {1+x^4}}+\frac {3 \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 {3 \left (5+5 \sqrt {5}\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 )}{20 \left (1+\sqrt {5}\right ) \sqrt {1+x^4}}+2 \left (\frac {3}{10} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\right )+\frac {1}{20} \left (3 \left (1-\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{25} \left (2 \left (5-\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{50} \left (9 \left (5-\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {2 \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx}{5 \left (-3+\sqrt {5}\right )}+\frac {\left (-5+\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{5 \left (-3+\sqrt {5}\right )}+\frac {\left (2 \left (-5+\sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-3+\sqrt {5}\right )}-\frac {\left (-1+\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{5 \left (-3+\sqrt {5}\right )}+\frac {1}{20} \left (3 \left (1+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {2 \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx}{5 \left (3+\sqrt {5}\right )}-\frac {\left (1+\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{5 \left (3+\sqrt {5}\right )}+\frac {1}{25} \left (2 \left (5+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{50} \left (9 \left (5+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {\left (5+\sqrt {5}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{5 \left (3+\sqrt {5}\right )}-\frac {\left (2 \left (5+\sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (3+\sqrt {5}\right )}+\frac {\left (3 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{10 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}+\frac {\left (12 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{25 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}+\frac {\left (3 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}+\frac {\left (24 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{25 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}-\frac {\left (27 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{25 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}-\frac {\left (54 \left (5-3 \sqrt {5}\right ) \left (-5+\sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (-3+\sqrt {5}-2 x^2\right ) \sqrt {1+x^4}} \, dx}{25 \left (-4+\left (-3+\sqrt {5}\right )^2\right )}-\frac {\left (3 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{10 \left (-4+\left (3+\sqrt {5}\right )^2\right )}-\frac {\left (12 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{25 \left (-4+\left (3+\sqrt {5}\right )^2\right )}+\frac {\left (3 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{5 \left (-4+\left (3+\sqrt {5}\right )^2\right )}+\frac {\left (24 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{25 \left (-4+\left (3+\sqrt {5}\right )^2\right )}+\frac {\left (27 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{25 \left (-4+\left (3+\sqrt {5}\right )^2\right )}-\frac {\left (54 \left (5+\sqrt {5}\right ) \left (5+3 \sqrt {5}\right )\right ) \int \frac {1+x^2}{\left (3+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx}{25 \left (-4+\left (3+\sqrt {5}\right )^2\right )}\\ &=\frac {2 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}\right ) \left (1+x^2\right )}+\frac {2 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}\right ) \left (1+x^2\right )}+\frac {3 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}+2 x^2\right )}-\frac {4 x \sqrt {1+x^4}}{5 \left (3-\sqrt {5}\right ) \left (3-\sqrt {5}+2 x^2\right )}+\frac {3 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}+2 x^2\right )}-\frac {4 x \sqrt {1+x^4}}{5 \left (3+\sqrt {5}\right ) \left (3+\sqrt {5}+2 x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{\sqrt {15} \left (3-\sqrt {5}\right )}-\frac {\sqrt {3} \left (1-\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{4 \left (5-\sqrt {5}\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{\sqrt {15} \left (3+\sqrt {5}\right )}-\frac {\sqrt {3} \left (1+\sqrt {5}\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{4 \left (5+\sqrt {5}\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 )}{5 \left (3-\sqrt {5}\right ) \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 )}{5 \left (3+\sqrt {5}\right ) \sqrt {1+x^4}}+2 \left (-\frac {3 x \sqrt {1+x^4}}{10 \left (1+x^2\right )}+\frac {3 \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 )}{10 \sqrt {1+x^4}}\right )+\frac {3 \left (1-\sqrt {5}\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 )}{40 \sqrt {1+x^4}}-\frac {\left (1-\sqrt {5}\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 )}{10 \left (3-\sqrt {5}\right ) \sqrt {1+x^4}}-\frac {3 \left (3-\sqrt {5}\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 (5-\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (5-\sqrt {5}\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 )}{10 \left (3-\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {3 \left (1+\sqrt {5}\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 )}{40 \sqrt {1+x^4}}-\frac {\left (1+\sqrt {5}\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 )}{10 \left (3+\sqrt {5}\right ) \sqrt {1+x^4}}-\frac {3 \left (3+\sqrt {5}\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 (5+\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {\left (5+\sqrt {5}\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 )}{10 \left (3+\sqrt {5}\right ) \sqrt {1+x^4}}+\frac {3 \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 (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 \left (3-\sqrt {5}\right ) \sqrt {1+x^4}}-\frac {\left (5+\sqrt {5}\right )^2 \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 )}{20 \left (3+\sqrt {5}\right )^2 \sqrt {1+x^4}}+\frac {3 \left (5+5 \sqrt {5}\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 )}{40 \left (1+\sqrt {5}\right ) \sqrt {1+x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.37, size = 257, normalized size = 4.85 \begin {gather*} -\frac {x^5-\sqrt [4]{-1} \sqrt {x^4+1} x^4 \Pi \left (-\frac {2 i}{3+\sqrt {5}};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-\sqrt [4]{-1} \sqrt {x^4+1} \Pi \left (-\frac {2 i}{3+\sqrt {5}};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\sqrt [4]{-1} \sqrt {x^4+1} \left (x^4+3 x^2+1\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-3 \sqrt [4]{-1} \sqrt {x^4+1} x^2 \Pi \left (-\frac {2 i}{3+\sqrt {5}};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-\sqrt [4]{-1} \sqrt {x^4+1} \left (x^4+3 x^2+1\right ) \Pi \left (\frac {2 i}{-3+\sqrt {5}};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+x}{2 \sqrt {x^4+1} \left (x^4+3 x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.63, size = 53, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {1+x^4}}{2 \left (1+3 x^2+x^4\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^4}}\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 65, normalized size = 1.23 \begin {gather*} -\frac {\sqrt {3} {\left (x^{4} + 3 \, x^{2} + 1\right )} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 1} x}{x^{4} - 3 \, x^{2} + 1}\right ) + 6 \, \sqrt {x^{4} + 1} x}{12 \, {\left (x^{4} + 3 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}{{\left (x^{4} + 3 \, x^{2} + 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.38, size = 60, normalized size = 1.13
method | result | size |
elliptic | \(\frac {\left (-\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{4 x \left (\frac {x^{4}+1}{2 x^{2}}+\frac {3}{2}\right )}+\frac {\sqrt {6}\, \arctan \left (\frac {\sqrt {6}\, \sqrt {2}\, \sqrt {x^{4}+1}}{6 x}\right )}{6}\right ) \sqrt {2}}{2}\) | \(60\) |
trager | \(-\frac {x \sqrt {x^{4}+1}}{2 \left (x^{4}+3 x^{2}+1\right )}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}-3 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}+6 \sqrt {x^{4}+1}\, x +\RootOf \left (\textit {\_Z}^{2}+3\right )}{x^{4}+3 x^{2}+1}\right )}{12}\) | \(85\) |
risch | \(-\frac {x \sqrt {x^{4}+1}}{2 \left (x^{4}+3 x^{2}+1\right )}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\sqrt {3}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-3\right )}{\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {6 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-3 \underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \underline {\hspace {1.25 ex}}\alpha ^{2}+3 i, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{24}\) | \(202\) |
default | \(-\frac {x \sqrt {x^{4}+1}}{2 \left (x^{4}+3 x^{2}+1\right )}+\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 {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (12 \underline {\hspace {1.25 ex}}\alpha ^{2}+23\right ) \left (-\frac {\sqrt {3}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-3\right )}{\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {6 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-3 \underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \underline {\hspace {1.25 ex}}\alpha ^{2}+3 i, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{120}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+3\right ) \left (-\frac {\sqrt {3}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-3\right )}{\sqrt {-3 \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {6 \left (-1\right )^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-3 \underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \underline {\hspace {1.25 ex}}\alpha ^{2}+3 i, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{20}\) | \(334\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + 1} {\left (x^{4} - 1\right )}}{{\left (x^{4} + 3 \, x^{2} + 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (x^4-1\right )\,\sqrt {x^4+1}}{{\left (x^4+3\,x^2+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________