Integrand size = 48, antiderivative size = 121 \[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\frac {\sqrt {4-3 x^2+x^4}}{2 x}+4 \text {arctanh}\left (\frac {x}{-2+x^2+\sqrt {4-3 x^2+x^4}}\right )-\frac {5}{2} \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} x}{-4+x+2 x^2+2 \sqrt {4-3 x^2+x^4}}\right )+\frac {5 \log (x)}{4}-\frac {5}{4} \log \left (-2+x^2+\sqrt {4-3 x^2+x^4}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.27 (sec) , antiderivative size = 917, normalized size of antiderivative = 7.58, number of steps used = 53, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6857, 1131, 1211, 1117, 1209, 1128, 748, 857, 633, 221, 738, 212, 1222, 1224, 1712, 213, 6860, 1742, 1230, 1720, 1261} \[ \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=\frac {5}{64} \left (7+\sqrt {33}\right ) \text {arcsinh}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+\frac {5}{64} \left (7-\sqrt {33}\right ) \text {arcsinh}\left (\frac {3-2 x^2}{\sqrt {7}}\right )-\frac {15}{32} \text {arcsinh}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt {x^4-3 x^2+4}}\right )-\frac {5}{4} \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} x}{2 \sqrt {x^4-3 x^2+4}}\right )+\frac {5}{8} \text {arctanh}\left (\frac {8-3 x^2}{4 \sqrt {x^4-3 x^2+4}}\right )-\frac {5}{8} \sqrt {5} \text {arctanh}\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} \text {arctanh}\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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticPi}\left (\frac {33}{32},2 \arctan \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}} \operatorname {EllipticPi}\left (\frac {33}{32},2 \arctan \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} \]
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Rule 212
Rule 213
Rule 221
Rule 633
Rule 738
Rule 748
Rule 857
Rule 1117
Rule 1128
Rule 1131
Rule 1209
Rule 1211
Rule 1222
Rule 1224
Rule 1230
Rule 1261
Rule 1712
Rule 1720
Rule 1742
Rule 6857
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \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} \text {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} \text {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 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {7}{8}\right )}{\sqrt {4-3 x^2+x^4}}+\frac {15}{32} \text {Subst}\left (\int \frac {1}{\sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-\frac {5}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {4-3 x+x^2}} \, dx,x,x^2\right )-4 \text {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 \text {arctanh}\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 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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} \text {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 \text {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 ) \text {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 ) \text {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} \text {arcsinh}\left (\frac {3-2 x^2}{\sqrt {7}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt {4-3 x^2+x^4}}\right )+\frac {5}{8} \text {arctanh}\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 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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}} \operatorname {EllipticF}\left (2 \arctan \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 ) \text {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 ) \text {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 )} \\ & = \text {Too large to display} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.97 \[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\frac {1}{4} \left (\frac {2 \sqrt {4-3 x^2+x^4}}{x}+16 \text {arctanh}\left (\frac {x}{-2+x^2+\sqrt {4-3 x^2+x^4}}\right )-10 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {5} x}{-4+x+2 x^2+2 \sqrt {4-3 x^2+x^4}}\right )+5 \log (x)-5 \log \left (-2+x^2+\sqrt {4-3 x^2+x^4}\right )\right ) \]
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Time = 2.79 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {\sqrt {x^{4}-3 x^{2}+4}}{2 x}+2 \,\operatorname {arctanh}\left (\frac {x}{\sqrt {x^{4}-3 x^{2}+4}}\right )-\frac {5 \,\operatorname {arcsinh}\left (\frac {x^{2}-2}{x}\right )}{4}+\frac {5 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-2 x -2\right ) \sqrt {5}}{5 \sqrt {x^{4}-3 x^{2}+4}}\right )}{4}\) | \(79\) |
default | \(\frac {5 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-2 x -2\right ) \sqrt {5}}{5 \sqrt {x^{4}-3 x^{2}+4}}\right ) x +8 \,\operatorname {arctanh}\left (\frac {x}{\sqrt {x^{4}-3 x^{2}+4}}\right ) x -5 \,\operatorname {arcsinh}\left (\frac {x^{2}-2}{x}\right ) x +2 \sqrt {x^{4}-3 x^{2}+4}}{4 x}\) | \(84\) |
pseudoelliptic | \(\frac {5 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-2 x -2\right ) \sqrt {5}}{5 \sqrt {x^{4}-3 x^{2}+4}}\right ) x +8 \,\operatorname {arctanh}\left (\frac {x}{\sqrt {x^{4}-3 x^{2}+4}}\right ) x -5 \,\operatorname {arcsinh}\left (\frac {x^{2}-2}{x}\right ) x +2 \sqrt {x^{4}-3 x^{2}+4}}{4 x}\) | \(84\) |
elliptic | \(-\frac {5 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {7}\, \left (x^{2}-\frac {3}{2}\right )}{7}\right )}{8}-\frac {160 \,\operatorname {arctanh}\left (\frac {-3 x^{2}+8}{4 \sqrt {x^{4}-3 x^{2}+4}}\right )}{\left (17+\sqrt {33}\right ) \left (-17+\sqrt {33}\right )}-\frac {25 \left (-297+25 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {\frac {85}{4}-\frac {5 \sqrt {33}}{4}+4 \left (\frac {5}{4}-\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}+\frac {\sqrt {33}}{8}\right )}{\left (\frac {\sqrt {165}}{8}-\frac {\sqrt {5}}{8}\right ) \sqrt {64 \left (x^{2}-\frac {17}{8}+\frac {\sqrt {33}}{8}\right )^{2}+64 \left (\frac {5}{4}-\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}+\frac {\sqrt {33}}{8}\right )+170-10 \sqrt {33}}}\right )}{16 \left (-1122+66 \sqrt {33}\right ) \left (\frac {\sqrt {165}}{8}-\frac {\sqrt {5}}{8}\right )}+\frac {25 \left (297+25 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {\frac {85}{4}+\frac {5 \sqrt {33}}{4}+4 \left (\frac {5}{4}+\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}-\frac {\sqrt {33}}{8}\right )}{\left (\frac {\sqrt {165}}{8}+\frac {\sqrt {5}}{8}\right ) \sqrt {64 \left (x^{2}-\frac {17}{8}-\frac {\sqrt {33}}{8}\right )^{2}+64 \left (\frac {5}{4}+\frac {\sqrt {33}}{4}\right ) \left (x^{2}-\frac {17}{8}-\frac {\sqrt {33}}{8}\right )+170+10 \sqrt {33}}}\right )}{16 \left (1122+66 \sqrt {33}\right ) \left (\frac {\sqrt {165}}{8}+\frac {\sqrt {5}}{8}\right )}+\frac {\left (\frac {\sqrt {x^{4}-3 x^{2}+4}\, \sqrt {2}}{2 x}-\frac {5 \sqrt {10}\, \operatorname {arctanh}\left (\frac {\sqrt {10}\, \sqrt {x^{4}-3 x^{2}+4}\, \sqrt {2}}{5 x}\right )}{4}+2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}-3 x^{2}+4}}{x}\right )\right ) \sqrt {2}}{2}\) | \(364\) |
trager | \(\frac {\sqrt {x^{4}-3 x^{2}+4}}{2 x}+\frac {\ln \left (\frac {131072+524288 x -30198 x^{20}+4373760 x^{10}+1296128 x^{9}-2095616 x^{7}+65536 \sqrt {x^{4}-3 x^{2}+4}+154096 x^{18}+1437984 x^{14}+9128 x^{21}+3507 x^{22}-32744 x^{19}-156064 x^{15}-1568 x^{23}+81008 x^{17}+2336768 x^{5}+3865344 x^{6}+237568 x^{2}-1605632 x^{3}-4931072 x^{8}-1795584 x^{4}-546720 x^{16}+292096 x^{13}-2875968 x^{12}-624256 x^{11}-16 x^{26}+128 x^{25}-116 x^{24}+16 \sqrt {x^{4}-3 x^{2}+4}\, x^{24}-128 \sqrt {x^{4}-3 x^{2}+4}\, x^{23}+140 \sqrt {x^{4}-3 x^{2}+4}\, x^{22}+1376 \sqrt {x^{4}-3 x^{2}+4}\, x^{21}-3311 \sqrt {x^{4}-3 x^{2}+4}\, x^{20}-6952 \sqrt {x^{4}-3 x^{2}+4}\, x^{19}+25088 \sqrt {x^{4}-3 x^{2}+4}\, x^{18}+21280 \sqrt {x^{4}-3 x^{2}+4}\, x^{17}-113776 \sqrt {x^{4}-3 x^{2}+4}\, x^{16}-44608 \sqrt {x^{4}-3 x^{2}+4}\, x^{15}+358208 \sqrt {x^{4}-3 x^{2}+4}\, x^{14}+77632 \sqrt {x^{4}-3 x^{2}+4}\, x^{13}-828960 \sqrt {x^{4}-3 x^{2}+4}\, x^{12}-155264 \sqrt {x^{4}-3 x^{2}+4}\, x^{11}+1432832 \sqrt {x^{4}-3 x^{2}+4}\, x^{10}+356864 \sqrt {x^{4}-3 x^{2}+4}\, x^{9}-1820416 \sqrt {x^{4}-3 x^{2}+4}\, x^{8}-680960 \sqrt {x^{4}-3 x^{2}+4}\, x^{7}+1605632 \sqrt {x^{4}-3 x^{2}+4}\, x^{6}+889856 \sqrt {x^{4}-3 x^{2}+4}\, x^{5}-847616 \sqrt {x^{4}-3 x^{2}+4}\, x^{4}-704512 x^{3} \sqrt {x^{4}-3 x^{2}+4}+143360 \sqrt {x^{4}-3 x^{2}+4}\, x^{2}+262144 x \sqrt {x^{4}-3 x^{2}+4}}{\left (x^{2}-2\right )^{8} x^{5}}\right )}{4}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )-5 \sqrt {x^{4}-3 x^{2}+4}}{2 x^{2}+x -4}\right )}{4}\) | \(645\) |
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Time = 0.38 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.24 \[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\frac {5 \, \sqrt {5} x \log \left (\frac {6 \, x^{4} - 4 \, x^{3} + 2 \, \sqrt {5} \sqrt {x^{4} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} - 15 \, x^{2} + 8 \, x + 24}{4 \, x^{4} + 4 \, x^{3} - 15 \, x^{2} - 8 \, x + 16}\right ) + 16 \, x \log \left (-\frac {x + \sqrt {x^{4} - 3 \, x^{2} + 4}}{x^{2} - 2}\right ) + 10 \, x \log \left (-\frac {x^{2} - \sqrt {x^{4} - 3 \, x^{2} + 4} - 2}{x}\right ) + 4 \, \sqrt {x^{4} - 3 \, x^{2} + 4}}{8 \, x} \]
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\[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\int \frac {\left (x^{2} + 2\right ) \left (x^{2} - 2 x - 2\right ) \sqrt {x^{4} - 3 x^{2} + 4}}{x^{2} \left (x^{2} - 2\right ) \left (2 x^{2} + x - 4\right )}\, dx \]
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\[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\int { \frac {\sqrt {x^{4} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} {\left (x^{2} + 2\right )}}{{\left (2 \, x^{2} + x - 4\right )} {\left (x^{2} - 2\right )} x^{2}} \,d x } \]
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\[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\int { \frac {\sqrt {x^{4} - 3 \, x^{2} + 4} {\left (x^{2} - 2 \, x - 2\right )} {\left (x^{2} + 2\right )}}{{\left (2 \, x^{2} + x - 4\right )} {\left (x^{2} - 2\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (2+x^2\right ) \left (-2-2 x+x^2\right ) \sqrt {4-3 x^2+x^4}}{x^2 \left (-2+x^2\right ) \left (-4+x+2 x^2\right )} \, dx=\int -\frac {\left (x^2+2\right )\,\left (-x^2+2\,x+2\right )\,\sqrt {x^4-3\,x^2+4}}{x^2\,\left (x^2-2\right )\,\left (2\,x^2+x-4\right )} \,d x \]
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