Optimal. Leaf size=59 \[ -\frac {\sqrt {2 x^4+3 x^2+1} x}{2 \left (2 x^4+2 x^2+1\right )}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {2 x^4+3 x^2+1}}\right ) \]
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Rubi [C] time = 2.43, antiderivative size = 253, normalized size of antiderivative = 4.29, number of steps used = 90, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6742, 1226, 1189, 1100, 1136, 1214, 1456, 539, 1208, 6728} \begin {gather*} -\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)}-\frac {\left (x^2+1\right ) \sqrt {\frac {2 x^2+1}{x^2+1}} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{2 \sqrt {2 x^4+3 x^2+1}}+\frac {i \left (x^2+1\right ) \Pi \left (\frac {1}{2}-\frac {i}{2};\tan ^{-1}\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 ) \Pi \left (\frac {1}{2}+\frac {i}{2};\tan ^{-1}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 539
Rule 1100
Rule 1136
Rule 1189
Rule 1208
Rule 1214
Rule 1226
Rule 1456
Rule 6728
Rule 6742
Rubi steps
\begin {align*} \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 \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 ) \Pi \left (\frac {1}{2}-\frac {i}{2};\tan ^{-1}\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 ) \Pi \left (\frac {1}{2}+\frac {i}{2};\tan ^{-1}\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\left (\left .\tan ^{-1}(x)\right |-1\right )}{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\left (\left .\tan ^{-1}(x)\right |-1\right )}{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}} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{2 \sqrt {1+3 x^2+2 x^4}}+\frac {\left (1+x^2\right ) \Pi \left (\frac {1}{2}-\frac {i}{2};\tan ^{-1}\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 ) \Pi \left (\frac {1}{2}+\frac {i}{2};\tan ^{-1}\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\left (\left .\tan ^{-1}(x)\right |-1\right )}{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\left (\left .\tan ^{-1}(x)\right |-1\right )}{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}} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{2 \sqrt {1+3 x^2+2 x^4}}+\frac {i \left (1+x^2\right ) \Pi \left (\frac {1}{2}-\frac {i}{2};\tan ^{-1}\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 ) \Pi \left (\frac {1}{2}+\frac {i}{2};\tan ^{-1}\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*}
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Mathematica [C] time = 0.87, size = 199, normalized size = 3.37 \begin {gather*} \frac {-i \sqrt {2} \sqrt {x^2+1} \sqrt {2 x^2+1} F\left (i \sinh ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )+i \sqrt {2} \sqrt {x^2+1} \sqrt {2 x^2+1} \Pi \left (\frac {1}{2}-\frac {i}{2};i \sinh ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )+i \sqrt {2} \sqrt {x^2+1} \sqrt {2 x^2+1} \Pi \left (\frac {1}{2}+\frac {i}{2};i \sinh ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )-\frac {2 x \left (2 x^4+3 x^2+1\right )}{2 x^4+2 x^2+1}}{4 \sqrt {2 x^4+3 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.64, size = 59, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {1+3 x^2+2 x^4}}{2 \left (1+2 x^2+2 x^4\right )}-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {1+3 x^2+2 x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 92, normalized size = 1.56 \begin {gather*} \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 )}} \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 {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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 74, normalized size = 1.25
method | result | size |
elliptic | \(\frac {\left (-\frac {\sqrt {2 x^{4}+3 x^{2}+1}\, \sqrt {2}}{4 x \left (\frac {2 x^{4}+3 x^{2}+1}{2 x^{2}}-\frac {1}{2}\right )}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {2 x^{4}+3 x^{2}+1}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(74\) |
trager | \(-\frac {x \sqrt {2 x^{4}+3 x^{2}+1}}{2 \left (2 x^{4}+2 x^{2}+1\right )}-\frac {\ln \left (-\frac {2 x^{4}+2 \sqrt {2 x^{4}+3 x^{2}+1}\, x +4 x^{2}+1}{2 x^{4}+2 x^{2}+1}\right )}{4}\) | \(81\) |
risch | \(-\frac {x \sqrt {2 x^{4}+3 x^{2}+1}}{2 \left (2 x^{4}+2 x^{2}+1\right )}-\frac {i \sqrt {x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticF \left (i x , \sqrt {2}\right )}{2 \sqrt {2 x^{4}+3 x^{2}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2}+3\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+5 x^{2}+4\right )}{10 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {2 x^{4}+3 x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {4 i \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (i x , 2 \underline {\hspace {1.25 ex}}\alpha ^{2}+2, i \sqrt {-2}\right )}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )\right )}{8}\) | \(201\) |
default | \(-\frac {i \sqrt {x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticF \left (i x , \sqrt {2}\right )}{2 \sqrt {2 x^{4}+3 x^{2}+1}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-\frac {\arctanh \left (\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2}+3\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+5 x^{2}+4\right )}{10 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {2 x^{4}+3 x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {4 i \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (i x , 2 \underline {\hspace {1.25 ex}}\alpha ^{2}+2, i \sqrt {-2}\right )}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )\right )}{8}-\frac {x \sqrt {2 x^{4}+3 x^{2}+1}}{2 \left (2 x^{4}+2 x^{2}+1\right )}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-\frac {\arctanh \left (\frac {\left (4 \underline {\hspace {1.25 ex}}\alpha ^{2}+3\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+5 x^{2}+4\right )}{10 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {2 x^{4}+3 x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {4 i \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (i x , 2 \underline {\hspace {1.25 ex}}\alpha ^{2}+2, i \sqrt {-2}\right )}{\sqrt {2 x^{4}+3 x^{2}+1}}\right )\right )}{4}\) | \(340\) |
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 {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} \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 = -0.02 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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