Integrand size = 38, antiderivative size = 59 \[ \int \frac {\left (-1+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2} \, dx=-\frac {x \sqrt {1+3 x^2+2 x^4}}{2 \left (1+2 x^2+2 x^4\right )}-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {1+3 x^2+2 x^4}}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.69 (sec) , antiderivative size = 253, normalized size of antiderivative = 4.29, number of steps used = 90, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6874, 1240, 1203, 1114, 1150, 1228, 1470, 553, 1222, 6860} \[ \int \frac {\left (-1+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2} \, dx=-\frac {\left (x^2+1\right ) \sqrt {\frac {2 x^2+1}{x^2+1}} \operatorname {EllipticF}(\arctan (x),-1)}{2 \sqrt {2 x^4+3 x^2+1}}+\frac {i \left (x^2+1\right ) \operatorname {EllipticPi}\left (\frac {1}{2}-\frac {i}{2},\arctan \left (\sqrt {2} x\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {\frac {x^2+1}{2 x^2+1}} \sqrt {2 x^4+3 x^2+1}}-\frac {i \left (x^2+1\right ) \operatorname {EllipticPi}\left (\frac {1}{2}+\frac {i}{2},\arctan \left (\sqrt {2} x\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {\frac {x^2+1}{2 x^2+1}} \sqrt {2 x^4+3 x^2+1}}-\frac {i \sqrt {2 x^4+3 x^2+1} x}{-4 x^2-(2-2 i)}-\frac {i \sqrt {2 x^4+3 x^2+1} x}{4 x^2+(2+2 i)} \]
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Rule 553
Rule 1114
Rule 1150
Rule 1203
Rule 1222
Rule 1228
Rule 1240
Rule 1470
Rule 6860
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \left (1+x^2\right ) \sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2}+\frac {\sqrt {1+3 x^2+2 x^4}}{1+2 x^2+2 x^4}\right ) \, dx \\ & = -\left (2 \int \frac {\left (1+x^2\right ) \sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2} \, dx\right )+\int \frac {\sqrt {1+3 x^2+2 x^4}}{1+2 x^2+2 x^4} \, dx \\ & = -\left (2 \int \left (\frac {\sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2}+\frac {x^2 \sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2}\right ) \, dx\right )+\int \left (\frac {2 i \sqrt {1+3 x^2+2 x^4}}{(-2+2 i)-4 x^2}+\frac {2 i \sqrt {1+3 x^2+2 x^4}}{(2+2 i)+4 x^2}\right ) \, dx \\ & = 2 i \int \frac {\sqrt {1+3 x^2+2 x^4}}{(-2+2 i)-4 x^2} \, dx+2 i \int \frac {\sqrt {1+3 x^2+2 x^4}}{(2+2 i)+4 x^2} \, dx-2 \int \frac {\sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2} \, dx-2 \int \frac {x^2 \sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2} \, dx \\ & = (-1-i) \int \frac {1}{\left ((-2+2 i)-4 x^2\right ) \sqrt {1+3 x^2+2 x^4}} \, dx-\frac {1}{8} i \int \frac {(-8+4 i)-8 x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx-\frac {1}{8} i \int \frac {(8+4 i)+8 x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx+(1-i) \int \frac {1}{\left ((2+2 i)+4 x^2\right ) \sqrt {1+3 x^2+2 x^4}} \, dx-2 \int \left (\frac {(2-2 i) \sqrt {1+3 x^2+2 x^4}}{\left ((-2+2 i)-4 x^2\right )^2}-\frac {i \sqrt {1+3 x^2+2 x^4}}{(-2+2 i)-4 x^2}+\frac {(2+2 i) \sqrt {1+3 x^2+2 x^4}}{\left ((2+2 i)+4 x^2\right )^2}-\frac {i \sqrt {1+3 x^2+2 x^4}}{(2+2 i)+4 x^2}\right ) \, dx-2 \int \left (-\frac {4 \sqrt {1+3 x^2+2 x^4}}{\left ((-2+2 i)-4 x^2\right )^2}+\frac {2 i \sqrt {1+3 x^2+2 x^4}}{(-2+2 i)-4 x^2}-\frac {4 \sqrt {1+3 x^2+2 x^4}}{\left ((2+2 i)+4 x^2\right )^2}+\frac {2 i \sqrt {1+3 x^2+2 x^4}}{(2+2 i)+4 x^2}\right ) \, dx \\ & = \left (-\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx+\left (-\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx+\left (-\frac {1}{2}+\frac {i}{2}\right ) \int \frac {2+4 x^2}{\left ((-2+2 i)-4 x^2\right ) \sqrt {1+3 x^2+2 x^4}} \, dx-\left (-\frac {1}{2}-i\right ) \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx-\left (-\frac {1}{2}+i\right ) \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx+2 i \int \frac {\sqrt {1+3 x^2+2 x^4}}{(-2+2 i)-4 x^2} \, dx+2 i \int \frac {\sqrt {1+3 x^2+2 x^4}}{(2+2 i)+4 x^2} \, dx-4 i \int \frac {\sqrt {1+3 x^2+2 x^4}}{(-2+2 i)-4 x^2} \, dx-4 i \int \frac {\sqrt {1+3 x^2+2 x^4}}{(2+2 i)+4 x^2} \, dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {2+4 x^2}{\left ((2+2 i)+4 x^2\right ) \sqrt {1+3 x^2+2 x^4}} \, dx-(4-4 i) \int \frac {\sqrt {1+3 x^2+2 x^4}}{\left ((-2+2 i)-4 x^2\right )^2} \, dx-(4+4 i) \int \frac {\sqrt {1+3 x^2+2 x^4}}{\left ((2+2 i)+4 x^2\right )^2} \, dx+8 \int \frac {\sqrt {1+3 x^2+2 x^4}}{\left ((-2+2 i)-4 x^2\right )^2} \, dx+8 \int \frac {\sqrt {1+3 x^2+2 x^4}}{\left ((2+2 i)+4 x^2\right )^2} \, dx \\ & = -\frac {i x \sqrt {1+3 x^2+2 x^4}}{(-2+2 i)-4 x^2}-\frac {i x \sqrt {1+3 x^2+2 x^4}}{(2+2 i)+4 x^2}-(-2-2 i) \int \frac {1}{\left ((-2+2 i)-4 x^2\right ) \sqrt {1+3 x^2+2 x^4}} \, dx+\left (-\frac {1}{8}-\frac {i}{8}\right ) \int \frac {(-2+2 i)+4 x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx-\frac {1}{8} i \int \frac {(-8+4 i)-8 x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx-\frac {1}{8} i \int \frac {(8+4 i)+8 x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx+\frac {1}{4} i \int \frac {(-8+4 i)-8 x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx+\frac {1}{4} i \int \frac {(8+4 i)+8 x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx-2 i \int \frac {1}{\left ((-2+2 i)-4 x^2\right ) \sqrt {1+3 x^2+2 x^4}} \, dx-2 i \int \frac {1}{\left ((2+2 i)+4 x^2\right ) \sqrt {1+3 x^2+2 x^4}} \, dx-\frac {1}{8} \int \frac {(2+2 i)-4 x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx+\frac {1}{8} \int \frac {(-2+2 i)+4 x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx+\left (\frac {1}{8}-\frac {i}{8}\right ) \int \frac {(2+2 i)-4 x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx-(2-2 i) \int \frac {1}{\left ((2+2 i)+4 x^2\right ) \sqrt {1+3 x^2+2 x^4}} \, dx+-\frac {\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {\frac {1}{2}+\frac {x^2}{2}} \sqrt {2+4 x^2}\right ) \int \frac {\sqrt {2+4 x^2}}{\left ((-2+2 i)-4 x^2\right ) \sqrt {\frac {1}{2}+\frac {x^2}{2}}} \, dx}{\sqrt {1+3 x^2+2 x^4}}+\frac {\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {\frac {1}{2}+\frac {x^2}{2}} \sqrt {2+4 x^2}\right ) \int \frac {\sqrt {2+4 x^2}}{\sqrt {\frac {1}{2}+\frac {x^2}{2}} \left ((2+2 i)+4 x^2\right )} \, dx}{\sqrt {1+3 x^2+2 x^4}} \\ & = -\frac {i x \sqrt {1+3 x^2+2 x^4}}{(-2+2 i)-4 x^2}-\frac {i x \sqrt {1+3 x^2+2 x^4}}{(2+2 i)+4 x^2}+\frac {\left (1+x^2\right ) \operatorname {EllipticPi}\left (\frac {1}{2}-\frac {i}{2},\arctan \left (\sqrt {2} x\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {\frac {1+x^2}{1+2 x^2}} \sqrt {1+3 x^2+2 x^4}}+\frac {\left (1+x^2\right ) \operatorname {EllipticPi}\left (\frac {1}{2}+\frac {i}{2},\arctan \left (\sqrt {2} x\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {\frac {1+x^2}{1+2 x^2}} \sqrt {1+3 x^2+2 x^4}}-(-1-i) \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx-(-1+i) \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx-(-1+i) \int \frac {2+4 x^2}{\left ((-2+2 i)-4 x^2\right ) \sqrt {1+3 x^2+2 x^4}} \, dx+(-1-2 i) \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx+(-1+2 i) \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx+\left (-\frac {1}{2}-\frac {i}{2}\right ) \int \frac {x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx+\left (-\frac {1}{2}+\frac {i}{2}\right ) \int \frac {x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx-\left (-\frac {1}{2}-i\right ) \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx-\left (-\frac {1}{2}+i\right ) \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx+\left (-\frac {1}{4}+\frac {i}{4}\right ) \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx-\left (\frac {1}{4}+\frac {i}{4}\right ) \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx+2 \left (\frac {1}{2} \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx\right )+2 \left (\frac {1}{2} \int \frac {x^2}{\sqrt {1+3 x^2+2 x^4}} \, dx\right )-(1+i) \int \frac {2+4 x^2}{\left ((2+2 i)+4 x^2\right ) \sqrt {1+3 x^2+2 x^4}} \, dx-2 \int \frac {1}{\sqrt {1+3 x^2+2 x^4}} \, dx-\int \frac {2+4 x^2}{\left ((-2+2 i)-4 x^2\right ) \sqrt {1+3 x^2+2 x^4}} \, dx+\int \frac {2+4 x^2}{\left ((2+2 i)+4 x^2\right ) \sqrt {1+3 x^2+2 x^4}} \, dx \\ & = -\frac {x \left (1+2 x^2\right )}{2 \sqrt {1+3 x^2+2 x^4}}-\frac {i x \sqrt {1+3 x^2+2 x^4}}{(-2+2 i)-4 x^2}-\frac {i x \sqrt {1+3 x^2+2 x^4}}{(2+2 i)+4 x^2}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+2 x^2}{1+x^2}} E(\arctan (x)|-1)}{2 \sqrt {1+3 x^2+2 x^4}}+2 \left (\frac {x \left (1+2 x^2\right )}{4 \sqrt {1+3 x^2+2 x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+2 x^2}{1+x^2}} E(\arctan (x)|-1)}{4 \sqrt {1+3 x^2+2 x^4}}\right )-\frac {\left (1+x^2\right ) \sqrt {\frac {1+2 x^2}{1+x^2}} \operatorname {EllipticF}(\arctan (x),-1)}{2 \sqrt {1+3 x^2+2 x^4}}+\frac {\left (1+x^2\right ) \operatorname {EllipticPi}\left (\frac {1}{2}-\frac {i}{2},\arctan \left (\sqrt {2} x\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {\frac {1+x^2}{1+2 x^2}} \sqrt {1+3 x^2+2 x^4}}+\frac {\left (1+x^2\right ) \operatorname {EllipticPi}\left (\frac {1}{2}+\frac {i}{2},\arctan \left (\sqrt {2} x\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {\frac {1+x^2}{1+2 x^2}} \sqrt {1+3 x^2+2 x^4}}--\frac {\left ((1-i) \sqrt {\frac {1}{2}+\frac {x^2}{2}} \sqrt {2+4 x^2}\right ) \int \frac {\sqrt {2+4 x^2}}{\left ((-2+2 i)-4 x^2\right ) \sqrt {\frac {1}{2}+\frac {x^2}{2}}} \, dx}{\sqrt {1+3 x^2+2 x^4}}-\frac {\left (\sqrt {\frac {1}{2}+\frac {x^2}{2}} \sqrt {2+4 x^2}\right ) \int \frac {\sqrt {2+4 x^2}}{\left ((-2+2 i)-4 x^2\right ) \sqrt {\frac {1}{2}+\frac {x^2}{2}}} \, dx}{\sqrt {1+3 x^2+2 x^4}}+\frac {\left (\sqrt {\frac {1}{2}+\frac {x^2}{2}} \sqrt {2+4 x^2}\right ) \int \frac {\sqrt {2+4 x^2}}{\sqrt {\frac {1}{2}+\frac {x^2}{2}} \left ((2+2 i)+4 x^2\right )} \, dx}{\sqrt {1+3 x^2+2 x^4}}-\frac {\left ((1+i) \sqrt {\frac {1}{2}+\frac {x^2}{2}} \sqrt {2+4 x^2}\right ) \int \frac {\sqrt {2+4 x^2}}{\sqrt {\frac {1}{2}+\frac {x^2}{2}} \left ((2+2 i)+4 x^2\right )} \, dx}{\sqrt {1+3 x^2+2 x^4}} \\ & = -\frac {x \left (1+2 x^2\right )}{2 \sqrt {1+3 x^2+2 x^4}}-\frac {i x \sqrt {1+3 x^2+2 x^4}}{(-2+2 i)-4 x^2}-\frac {i x \sqrt {1+3 x^2+2 x^4}}{(2+2 i)+4 x^2}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+2 x^2}{1+x^2}} E(\arctan (x)|-1)}{2 \sqrt {1+3 x^2+2 x^4}}+2 \left (\frac {x \left (1+2 x^2\right )}{4 \sqrt {1+3 x^2+2 x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+2 x^2}{1+x^2}} E(\arctan (x)|-1)}{4 \sqrt {1+3 x^2+2 x^4}}\right )-\frac {\left (1+x^2\right ) \sqrt {\frac {1+2 x^2}{1+x^2}} \operatorname {EllipticF}(\arctan (x),-1)}{2 \sqrt {1+3 x^2+2 x^4}}+\frac {i \left (1+x^2\right ) \operatorname {EllipticPi}\left (\frac {1}{2}-\frac {i}{2},\arctan \left (\sqrt {2} x\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {\frac {1+x^2}{1+2 x^2}} \sqrt {1+3 x^2+2 x^4}}-\frac {i \left (1+x^2\right ) \operatorname {EllipticPi}\left (\frac {1}{2}+\frac {i}{2},\arctan \left (\sqrt {2} x\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {\frac {1+x^2}{1+2 x^2}} \sqrt {1+3 x^2+2 x^4}} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2} \, dx=-\frac {x \sqrt {1+3 x^2+2 x^4}}{2 \left (1+2 x^2+2 x^4\right )}-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {1+3 x^2+2 x^4}}\right ) \]
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Time = 5.48 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.25
method | result | size |
elliptic | \(\frac {\left (-\frac {\sqrt {2 x^{4}+3 x^{2}+1}\, \sqrt {2}}{4 x \left (\frac {2 x^{4}+3 x^{2}+1}{2 x^{2}}-\frac {1}{2}\right )}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2 x^{4}+3 x^{2}+1}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(74\) |
trager | \(-\frac {x \sqrt {2 x^{4}+3 x^{2}+1}}{2 \left (2 x^{4}+2 x^{2}+1\right )}+\frac {\ln \left (-\frac {-2 x^{4}+2 x \sqrt {2 x^{4}+3 x^{2}+1}-4 x^{2}-1}{2 x^{4}+2 x^{2}+1}\right )}{4}\) | \(81\) |
risch | \(-\frac {x \sqrt {2 x^{4}+3 x^{2}+1}}{2 \left (2 x^{4}+2 x^{2}+1\right )}-\frac {\operatorname {arctanh}\left (\frac {\left (2 x^{2}+\sqrt {2}\right ) \sqrt {\sqrt {2}-1}-2 x \sqrt {2}+3 x}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )}{4}+\frac {\operatorname {arctanh}\left (\frac {\left (2 x^{2}+\sqrt {2}\right ) \sqrt {\sqrt {2}-1}+2 x \sqrt {2}-3 x}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )}{4}\) | \(123\) |
default | \(\frac {\left (-2 x^{4}-2 x^{2}-1\right ) \operatorname {arctanh}\left (\frac {\left (2 x^{2}+\sqrt {2}\right ) \sqrt {\sqrt {2}-1}-2 x \sqrt {2}+3 x}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )+\left (2 x^{4}+2 x^{2}+1\right ) \operatorname {arctanh}\left (\frac {\left (2 x^{2}+\sqrt {2}\right ) \sqrt {\sqrt {2}-1}+2 x \sqrt {2}-3 x}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )-2 x \sqrt {2 x^{4}+3 x^{2}+1}}{-8 \left (\sqrt {2}-1\right ) x^{2}+2 \left (2 x^{2}+\sqrt {2}\right )^{2}}\) | \(158\) |
pseudoelliptic | \(\frac {\left (-2 x^{4}-2 x^{2}-1\right ) \operatorname {arctanh}\left (\frac {\left (2 x^{2}+\sqrt {2}\right ) \sqrt {\sqrt {2}-1}-2 x \sqrt {2}+3 x}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )+\left (2 x^{4}+2 x^{2}+1\right ) \operatorname {arctanh}\left (\frac {\left (2 x^{2}+\sqrt {2}\right ) \sqrt {\sqrt {2}-1}+2 x \sqrt {2}-3 x}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )-2 x \sqrt {2 x^{4}+3 x^{2}+1}}{-8 \left (\sqrt {2}-1\right ) x^{2}+2 \left (2 x^{2}+\sqrt {2}\right )^{2}}\) | \(158\) |
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Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.56 \[ \int \frac {\left (-1+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2} \, dx=\frac {{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )} \log \left (\frac {2 \, x^{4} + 4 \, x^{2} - 2 \, \sqrt {2 \, x^{4} + 3 \, x^{2} + 1} x + 1}{2 \, x^{4} + 2 \, x^{2} + 1}\right ) - 2 \, \sqrt {2 \, x^{4} + 3 \, x^{2} + 1} x}{4 \, {\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}} \]
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\[ \int \frac {\left (-1+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2} \, dx=\int \frac {\sqrt {\left (x^{2} + 1\right ) \left (2 x^{2} + 1\right )} \left (2 x^{4} - 1\right )}{\left (2 x^{4} + 2 x^{2} + 1\right )^{2}}\, dx \]
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\[ \int \frac {\left (-1+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2} \, dx=\int { \frac {\sqrt {2 \, x^{4} + 3 \, x^{2} + 1} {\left (2 \, x^{4} - 1\right )}}{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (-1+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2} \, dx=\int { \frac {\sqrt {2 \, x^{4} + 3 \, x^{2} + 1} {\left (2 \, x^{4} - 1\right )}}{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+2 x^4\right ) \sqrt {1+3 x^2+2 x^4}}{\left (1+2 x^2+2 x^4\right )^2} \, dx=\int \frac {\left (2\,x^4-1\right )\,\sqrt {2\,x^4+3\,x^2+1}}{{\left (2\,x^4+2\,x^2+1\right )}^2} \,d x \]
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