Optimal. Leaf size=121 \[ \frac {\sqrt {x^4-3 x^2+4}}{2 x}-\frac {5}{4} \log \left (x^2+\sqrt {x^4-3 x^2+4}-2\right )+4 \tanh ^{-1}\left (\frac {x}{x^2+\sqrt {x^4-3 x^2+4}-2}\right )-\frac {5}{2} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 x^2+2 \sqrt {x^4-3 x^2+4}+x-4}\right )+\frac {5 \log (x)}{4} \]
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Rubi [C] time = 3.65, antiderivative size = 917, normalized size of antiderivative = 7.58, number of steps used = 53, number of rules used = 21, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6725, 1117, 1197, 1103, 1195, 1114, 734, 843, 619, 215, 724, 206, 1208, 1210, 1698, 207, 6728, 1728, 1216, 1706, 1247} \begin {gather*} \frac {5}{64} \left (7+\sqrt {33}\right ) \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+\frac {5}{64} \left (7-\sqrt {33}\right ) \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )-\frac {15}{32} \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^4-3 x^2+4}}\right )-\frac {5}{4} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {x^4-3 x^2+4}}\right )+\frac {5}{8} \tanh ^{-1}\left (\frac {8-3 x^2}{4 \sqrt {x^4-3 x^2+4}}\right )-\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {2 \left (5-\sqrt {33}\right ) x^2+3 \sqrt {33}+13}{2 \sqrt {10 \left (17-\sqrt {33}\right )} \sqrt {x^4-3 x^2+4}}\right )+\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {2 \left (5+\sqrt {33}\right ) x^2-3 \sqrt {33}+13}{2 \sqrt {10 \left (17+\sqrt {33}\right )} \sqrt {x^4-3 x^2+4}}\right )-\frac {25 \left (17+\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33+\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {5 \left (9+\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {x^4-3 x^2+4}}-\frac {25 \left (17-\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33-\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {5 \left (9-\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {x^4-3 x^2+4}}-\frac {5 \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {x^4-3 x^2+4}}+\frac {25 \left (17+\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \left (33+17 \sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {25 \left (17-\sqrt {33}\right ) \left (x^2+2\right ) \sqrt {\frac {x^4-3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \left (33-17 \sqrt {33}\right ) \sqrt {x^4-3 x^2+4}}+\frac {\sqrt {x^4-3 x^2+4}}{2 x}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {x^4-3 x^2+4}-\frac {5}{16} \sqrt {x^4-3 x^2+4} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 206
Rule 207
Rule 215
Rule 619
Rule 724
Rule 734
Rule 843
Rule 1103
Rule 1114
Rule 1117
Rule 1195
Rule 1197
Rule 1208
Rule 1210
Rule 1216
Rule 1247
Rule 1698
Rule 1706
Rule 1728
Rule 6725
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx &=\int \left (-\frac {\sqrt {4-3 x^2+x^4}}{2 x^2}-\frac {5 \sqrt {4-3 x^2+x^4}}{8 x}-\frac {4 \sqrt {4-3 x^2+x^4}}{-2+x^2}+\frac {5 (17+2 x) \sqrt {4-3 x^2+x^4}}{8 \left (-4+x+2 x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt {4-3 x^2+x^4}}{x^2} \, dx\right )-\frac {5}{8} \int \frac {\sqrt {4-3 x^2+x^4}}{x} \, dx+\frac {5}{8} \int \frac {(17+2 x) \sqrt {4-3 x^2+x^4}}{-4+x+2 x^2} \, dx-4 \int \frac {\sqrt {4-3 x^2+x^4}}{-2+x^2} \, dx\\ &=\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5}{16} \operatorname {Subst}\left (\int \frac {\sqrt {4-3 x+x^2}}{x} \, dx,x,x^2\right )-\frac {1}{2} \int \frac {-3+2 x^2}{\sqrt {4-3 x^2+x^4}} \, dx+\frac {5}{8} \int \left (\frac {\left (2+2 \sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}{1-\sqrt {33}+4 x}+\frac {\left (2-2 \sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}{1+\sqrt {33}+4 x}\right ) \, dx+4 \int \frac {1-x^2}{\sqrt {4-3 x^2+x^4}} \, dx-8 \int \frac {1}{\left (-2+x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}+\frac {5}{32} \operatorname {Subst}\left (\int \frac {-8+3 x}{x \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-\frac {1}{2} \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx+2 \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx+2 \int \frac {1-\frac {x^2}{2}}{\sqrt {4-3 x^2+x^4}} \, dx+2 \int \frac {-2-x^2}{\left (-2+x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx-4 \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx+8 \int \frac {1-\frac {x^2}{2}}{\sqrt {4-3 x^2+x^4}} \, dx+\frac {1}{4} \left (5 \left (1-\sqrt {33}\right )\right ) \int \frac {\sqrt {4-3 x^2+x^4}}{1+\sqrt {33}+4 x} \, dx+\frac {1}{4} \left (5 \left (1+\sqrt {33}\right )\right ) \int \frac {\sqrt {4-3 x^2+x^4}}{1-\sqrt {33}+4 x} \, dx\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {15}{32} \operatorname {Subst}\left (\int \frac {1}{\sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-\frac {5}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-4 \operatorname {Subst}\left (\int \frac {1}{-2+2 x^2} \, dx,x,\frac {x}{\sqrt {4-3 x^2+x^4}}\right )-40 \int \frac {\sqrt {4-3 x^2+x^4}}{\left (1-\sqrt {33}\right )^2-16 x^2} \, dx-40 \int \frac {\sqrt {4-3 x^2+x^4}}{\left (1+\sqrt {33}\right )^2-16 x^2} \, dx-\left (5 \left (1-\sqrt {33}\right )\right ) \int \frac {x \sqrt {4-3 x^2+x^4}}{\left (1+\sqrt {33}\right )^2-16 x^2} \, dx-\left (5 \left (1+\sqrt {33}\right )\right ) \int \frac {x \sqrt {4-3 x^2+x^4}}{\left (1-\sqrt {33}\right )^2-16 x^2} \, dx\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}+2 \tanh ^{-1}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {5}{32} \int \frac {-48+\left (1-\sqrt {33}\right )^2+16 x^2}{\sqrt {4-3 x^2+x^4}} \, dx+\frac {5}{32} \int \frac {-48+\left (1+\sqrt {33}\right )^2+16 x^2}{\sqrt {4-3 x^2+x^4}} \, dx+\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,\frac {8-3 x^2}{\sqrt {4-3 x^2+x^4}}\right )+\frac {15 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,-3+2 x^2\right )}{32 \sqrt {7}}-\frac {1}{2} \left (5 \left (1-\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {4-3 x+x^2}}{\left (1+\sqrt {33}\right )^2-16 x} \, dx,x,x^2\right )-\frac {1}{4} \left (25 \left (17-\sqrt {33}\right )\right ) \int \frac {1}{\left (\left (1-\sqrt {33}\right )^2-16 x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx-\frac {1}{2} \left (5 \left (1+\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {4-3 x+x^2}}{\left (1-\sqrt {33}\right )^2-16 x} \, dx,x,x^2\right )-\frac {1}{4} \left (25 \left (17+\sqrt {33}\right )\right ) \int \frac {1}{\left (\left (1+\sqrt {33}\right )^2-16 x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}-\frac {15}{32} \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )+\frac {5}{8} \tanh ^{-1}\left (\frac {8-3 x^2}{4 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}-2 \left (5 \int \frac {1-\frac {x^2}{2}}{\sqrt {4-3 x^2+x^4}} \, dx\right )-\frac {1}{64} \left (5 \left (1-\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {2 \left (13-3 \sqrt {33}\right )+4 \left (5+\sqrt {33}\right ) x}{\left (\left (1+\sqrt {33}\right )^2-16 x\right ) \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )+\frac {1}{16} \left (5 \left (9-\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx-\frac {1}{64} \left (5 \left (1+\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {2 \left (13+3 \sqrt {33}\right )+4 \left (5-\sqrt {33}\right ) x}{\left (\left (1-\sqrt {33}\right )^2-16 x\right ) \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )+\frac {1}{16} \left (5 \left (9+\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx-\frac {\left (25 \left (17+\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx}{8 \left (33+\sqrt {33}\right )}-\frac {\left (100 \left (17+\sqrt {33}\right )\right ) \int \frac {1+\frac {x^2}{2}}{\left (\left (1+\sqrt {33}\right )^2-16 x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx}{33+\sqrt {33}}-\frac {\left (400 \left (17-\sqrt {33}\right ) \left (-16+\frac {1}{2} \left (1-\sqrt {33}\right )^2\right )\right ) \int \frac {1+\frac {x^2}{2}}{\left (\left (1-\sqrt {33}\right )^2-16 x^2\right ) \sqrt {4-3 x^2+x^4}} \, dx}{-1024+\left (1-\sqrt {33}\right )^4}-\frac {\left (25 \left (17-\sqrt {33}\right ) \left (-32+\left (1-\sqrt {33}\right )^2\right )\right ) \int \frac {1}{\sqrt {4-3 x^2+x^4}} \, dx}{4 \left (-1024+\left (1-\sqrt {33}\right )^4\right )}\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}-\frac {15}{32} \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )-\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {165} \left (17-\sqrt {33}\right ) \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {4-3 x^2+x^4}}\right )}{8 \left (33-17 \sqrt {33}\right )}+\frac {5}{8} \tanh ^{-1}\left (\frac {8-3 x^2}{4 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}-2 \left (-\frac {5 x \sqrt {4-3 x^2+x^4}}{2 \left (2+x^2\right )}+\frac {5 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {2} \sqrt {4-3 x^2+x^4}}\right )+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {5 \left (9-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {25 \left (17-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {5 \left (9+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {25 \left (17+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {25 \left (17+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \left (33+17 \sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {25 \left (17-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \left (33 \sqrt {2}-17 \sqrt {66}\right ) \sqrt {4-3 x^2+x^4}}+\frac {1}{4} \left (25 \left (1-\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1-\sqrt {33}\right )^2-16 x\right ) \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-\frac {1}{64} \left (5 \left (7-\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {4-3 x+x^2}} \, dx,x,x^2\right )+\frac {1}{4} \left (25 \left (1+\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+\sqrt {33}\right )^2-16 x\right ) \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-\frac {1}{64} \left (5 \left (7+\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {4-3 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}-\frac {15}{32} \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )-\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {165} \left (17-\sqrt {33}\right ) \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {4-3 x^2+x^4}}\right )}{8 \left (33-17 \sqrt {33}\right )}+\frac {5}{8} \tanh ^{-1}\left (\frac {8-3 x^2}{4 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}-2 \left (-\frac {5 x \sqrt {4-3 x^2+x^4}}{2 \left (2+x^2\right )}+\frac {5 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {2} \sqrt {4-3 x^2+x^4}}\right )+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {5 \left (9-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {25 \left (17-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {5 \left (9+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {25 \left (17+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {25 \left (17+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \left (33+17 \sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {25 \left (17-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \left (33 \sqrt {2}-17 \sqrt {66}\right ) \sqrt {4-3 x^2+x^4}}-\frac {1}{2} \left (25 \left (1-\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4096-192 \left (1-\sqrt {33}\right )^2+4 \left (1-\sqrt {33}\right )^4-x^2} \, dx,x,\frac {-128+3 \left (1-\sqrt {33}\right )^2-4 \left (5-\sqrt {33}\right ) x^2}{\sqrt {4-3 x^2+x^4}}\right )-\frac {\left (5 \left (7-\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,-3+2 x^2\right )}{64 \sqrt {7}}-\frac {1}{2} \left (25 \left (1+\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4096-192 \left (1+\sqrt {33}\right )^2+4 \left (1+\sqrt {33}\right )^4-x^2} \, dx,x,\frac {-128+3 \left (1+\sqrt {33}\right )^2-4 \left (5+\sqrt {33}\right ) x^2}{\sqrt {4-3 x^2+x^4}}\right )-\frac {\left (5 \left (7+\sqrt {33}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,-3+2 x^2\right )}{64 \sqrt {7}}\\ &=-\frac {5}{16} \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {5}{32} \left (1+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}+\frac {\sqrt {4-3 x^2+x^4}}{2 x}-\frac {5 x \sqrt {4-3 x^2+x^4}}{2+x^2}-\frac {15}{32} \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+\frac {5}{64} \left (7-\sqrt {33}\right ) \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+\frac {5}{64} \left (7+\sqrt {33}\right ) \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )-\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {165} \left (17-\sqrt {33}\right ) \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {4-3 x^2+x^4}}\right )}{8 \left (33-17 \sqrt {33}\right )}+\frac {5}{8} \tanh ^{-1}\left (\frac {8-3 x^2}{4 \sqrt {4-3 x^2+x^4}}\right )-\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {13+3 \sqrt {33}+2 \left (5-\sqrt {33}\right ) x^2}{2 \sqrt {10 \left (17-\sqrt {33}\right )} \sqrt {4-3 x^2+x^4}}\right )+\frac {5}{8} \sqrt {5} \tanh ^{-1}\left (\frac {13-3 \sqrt {33}+2 \left (5+\sqrt {33}\right ) x^2}{2 \sqrt {10 \left (17+\sqrt {33}\right )} \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}-2 \left (-\frac {5 x \sqrt {4-3 x^2+x^4}}{2 \left (2+x^2\right )}+\frac {5 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {2} \sqrt {4-3 x^2+x^4}}\right )+\frac {3 \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{4 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {\sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {5 \left (9-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {25 \left (17-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33-\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {5 \left (9+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \sqrt {4-3 x^2+x^4}}-\frac {25 \left (17+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{16 \sqrt {2} \left (33+\sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {25 \left (17+\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \sqrt {2} \left (33+17 \sqrt {33}\right ) \sqrt {4-3 x^2+x^4}}+\frac {25 \left (17-\sqrt {33}\right ) \left (2+x^2\right ) \sqrt {\frac {4-3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (\frac {33}{32};2 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )|\frac {7}{8}\right )}{32 \left (33 \sqrt {2}-17 \sqrt {66}\right ) \sqrt {4-3 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 1.60, size = 871, normalized size = 7.20 \begin {gather*} \frac {8 \sqrt {-\frac {i}{3 i+\sqrt {7}}} x^4-24 \sqrt {-\frac {i}{3 i+\sqrt {7}}} x^2+10 \sqrt {-\frac {i}{3 i+\sqrt {7}}} \sqrt {x^4-3 x^2+4} \sinh ^{-1}\left (\frac {3-2 x^2}{\sqrt {7}}\right ) x+10 \sqrt {-\frac {i}{3 i+\sqrt {7}}} \sqrt {x^4-3 x^2+4} \tanh ^{-1}\left (\frac {8-3 x^2}{4 \sqrt {x^4-3 x^2+4}}\right ) x+10 \sqrt {-\frac {5 i}{3 i+\sqrt {7}}} \sqrt {x^4-3 x^2+4} \tanh ^{-1}\left (\frac {2 \left (5+\sqrt {33}\right ) x^2-3 \sqrt {33}+13}{2 \sqrt {10 \left (17+\sqrt {33}\right )} \sqrt {x^4-3 x^2+4}}\right ) x-10 \sqrt {-\frac {5 i}{3 i+\sqrt {7}}} \sqrt {x^4-3 x^2+4} \tanh ^{-1}\left (\frac {-2 \left (-5+\sqrt {33}\right ) x^2+3 \sqrt {33}+13}{2 \sqrt {10} \sqrt {-\left (\left (-17+\sqrt {33}\right ) \left (x^4-3 x^2+4\right )\right )}}\right ) x-9 i \sqrt {\frac {2 i x^2}{-3 i+\sqrt {7}}+1} \sqrt {2-\frac {4 i x^2}{3 i+\sqrt {7}}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{3 i+\sqrt {7}}} x\right )|\frac {3 i+\sqrt {7}}{3 i-\sqrt {7}}\right ) x-32 i \sqrt {\frac {2 i x^2}{-3 i+\sqrt {7}}+1} \sqrt {2-\frac {4 i x^2}{3 i+\sqrt {7}}} \Pi \left (\frac {3}{4}-\frac {i \sqrt {7}}{4};i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{3 i+\sqrt {7}}} x\right )|\frac {3 i+\sqrt {7}}{3 i-\sqrt {7}}\right ) x+25 i \sqrt {\frac {2 i x^2}{-3 i+\sqrt {7}}+1} \sqrt {2-\frac {4 i x^2}{3 i+\sqrt {7}}} \Pi \left (\frac {4 i \left (3 i+\sqrt {7}\right )}{-17+\sqrt {33}};i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{3 i+\sqrt {7}}} x\right )|\frac {3 i+\sqrt {7}}{3 i-\sqrt {7}}\right ) x+25 i \sqrt {\frac {2 i x^2}{-3 i+\sqrt {7}}+1} \sqrt {2-\frac {4 i x^2}{3 i+\sqrt {7}}} \Pi \left (\frac {12-4 i \sqrt {7}}{17+\sqrt {33}};i \sinh ^{-1}\left (\sqrt {-\frac {2 i}{3 i+\sqrt {7}}} x\right )|\frac {3 i+\sqrt {7}}{3 i-\sqrt {7}}\right ) x+32 \sqrt {-\frac {i}{3 i+\sqrt {7}}}}{16 \sqrt {-\frac {i}{3 i+\sqrt {7}}} x \sqrt {x^4-3 x^2+4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.12, size = 121, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^4-3 x^2+4}}{2 x}-\frac {5}{4} \log \left (x^2+\sqrt {x^4-3 x^2+4}-2\right )+4 \tanh ^{-1}\left (\frac {x}{x^2+\sqrt {x^4-3 x^2+4}-2}\right )-\frac {5}{2} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5} x}{2 x^2+2 \sqrt {x^4-3 x^2+4}+x-4}\right )+\frac {5 \log (x)}{4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 150, normalized size = 1.24 \begin {gather*} \frac {5 \, \sqrt {5} x \log \left (\frac {6 \, x^{4} - 4 \, x^{3} + 2 \, \sqrt {5} \sqrt {x^{4} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} - 15 \, x^{2} + 8 \, x + 24}{4 \, x^{4} + 4 \, x^{3} - 15 \, x^{2} - 8 \, x + 16}\right ) + 16 \, x \log \left (-\frac {x + \sqrt {x^{4} - 3 \, x^{2} + 4}}{x^{2} - 2}\right ) + 10 \, x \log \left (-\frac {x^{2} - \sqrt {x^{4} - 3 \, x^{2} + 4} - 2}{x}\right ) + 4 \, \sqrt {x^{4} - 3 \, x^{2} + 4}}{8 \, x} \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} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} {\left (x^{2} + 2\right )}}{{\left (2 \, x^{2} + x - 4\right )} {\left (x^{2} - 2\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.66, size = 1643, normalized size = 13.58
result too large to display
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} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} {\left (x^{2} + 2\right )}}{{\left (2 \, x^{2} + x - 4\right )} {\left (x^{2} - 2\right )} x^{2}}\,{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+2\right )\,\left (-x^2+2\,x+2\right )\,\sqrt {x^4-3\,x^2+4}}{x^2\,\left (x^2-2\right )\,\left (2\,x^2+x-4\right )} \,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^{2} + 2\right ) \left (x^{2} - 2 x - 2\right ) \sqrt {x^{4} - 3 x^{2} + 4}}{x^{2} \left (x^{2} - 2\right ) \left (2 x^{2} + x - 4\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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