Optimal. Leaf size=186 \[ -4 \sqrt [4]{2} \text {RootSum}\left [1-4 \text {$\#$1}^2-122 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {-\log \left (-\sqrt {2}+2^{3/4} x-x^2\right ) \text {$\#$1}+\log \left (\sqrt {2+x^4}+\sqrt {2} \text {$\#$1}-2^{3/4} x \text {$\#$1}+x^2 \text {$\#$1}\right ) \text {$\#$1}-\log \left (-\sqrt {2}+2^{3/4} x-x^2\right ) \text {$\#$1}^3+\log \left (\sqrt {2+x^4}+\sqrt {2} \text {$\#$1}-2^{3/4} x \text {$\#$1}+x^2 \text {$\#$1}\right ) \text {$\#$1}^3}{-1-61 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 2.03, antiderivative size = 1539, normalized size of antiderivative = 8.27, number of steps
used = 20, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6860, 415,
226, 418, 1231, 1721} \begin {gather*} -\frac {\sqrt {1-i \sqrt {7}} \left (5 i+\sqrt {7}\right ) \text {ArcTan}\left (\frac {\sqrt {1-i \sqrt {7}} x}{\sqrt [4]{2 \left (-3-i \sqrt {7}\right )} \sqrt {x^4+2}}\right )}{2 \left (3 i-\sqrt {7}\right ) \left (2 \left (-3-i \sqrt {7}\right )\right )^{3/4}}-\frac {\left (1-3 i \sqrt {7}\right ) \text {ArcTan}\left (\frac {\sqrt {-1-i \sqrt {7}} x}{\sqrt [4]{2 \left (-3+i \sqrt {7}\right )} \sqrt {x^4+2}}\right )}{2^{3/4} \sqrt {-1-i \sqrt {7}} \left (-3+i \sqrt {7}\right )^{7/4}}-\frac {\left (1+3 i \sqrt {7}\right ) \text {ArcTan}\left (\frac {\sqrt {-1+i \sqrt {7}} x}{\sqrt [4]{2 \left (-3-i \sqrt {7}\right )} \sqrt {x^4+2}}\right )}{2^{3/4} \left (-3-i \sqrt {7}\right )^{7/4} \sqrt {-1+i \sqrt {7}}}-\frac {\left (5 i-\sqrt {7}\right ) \sqrt {1+i \sqrt {7}} \text {ArcTan}\left (\frac {\sqrt {1+i \sqrt {7}} x}{\sqrt [4]{2 \left (-3+i \sqrt {7}\right )} \sqrt {x^4+2}}\right )}{2 \left (2 \left (-3+i \sqrt {7}\right )\right )^{3/4} \left (3 i+\sqrt {7}\right )}+\frac {\left (1+i \sqrt {7}\right )^2 \left (1+\frac {2}{\sqrt {-3+i \sqrt {7}}}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7-i \sqrt {7}\right ) \sqrt {x^4+2}}+\frac {\left (1+i \sqrt {7}\right )^2 \left (1-\frac {2}{\sqrt {-3+i \sqrt {7}}}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7-i \sqrt {7}\right ) \sqrt {x^4+2}}+\frac {\left (1-i \sqrt {7}\right )^2 \left (1+\frac {2}{\sqrt {-3-i \sqrt {7}}}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7+i \sqrt {7}\right ) \sqrt {x^4+2}}+\frac {\left (1-i \sqrt {7}\right )^2 \left (1-\frac {2}{\sqrt {-3-i \sqrt {7}}}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7+i \sqrt {7}\right ) \sqrt {x^4+2}}+\frac {\left (1+i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {x^4+2}}+\frac {\left (1-i \sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left (3 i-\sqrt {7}\right ) \left (2+\sqrt {-3-i \sqrt {7}}\right )^2 \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \Pi \left (-\frac {\left (2-\sqrt {-3-i \sqrt {7}}\right )^2}{8 \sqrt {-3-i \sqrt {7}}};2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{16 \sqrt [4]{2} \left (7 i-5 \sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (3 i-\sqrt {7}\right ) \left (2-\sqrt {-3-i \sqrt {7}}\right )^2 \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \Pi \left (\frac {\left (2+\sqrt {-3-i \sqrt {7}}\right )^2}{8 \sqrt {-3-i \sqrt {7}}};2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{16 \sqrt [4]{2} \left (7 i-5 \sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (3 i+\sqrt {7}\right ) \left (2+\sqrt {-3+i \sqrt {7}}\right )^2 \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \Pi \left (-\frac {\left (2-\sqrt {-3+i \sqrt {7}}\right )^2}{8 \sqrt {-3+i \sqrt {7}}};2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{16 \sqrt [4]{2} \left (7 i+5 \sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (3 i+\sqrt {7}\right ) \left (2-\sqrt {-3+i \sqrt {7}}\right )^2 \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \Pi \left (\frac {\left (2+\sqrt {-3+i \sqrt {7}}\right )^2}{8 \sqrt {-3+i \sqrt {7}}};2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{16 \sqrt [4]{2} \left (7 i+5 \sqrt {7}\right ) \sqrt {x^4+2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 226
Rule 415
Rule 418
Rule 1231
Rule 1721
Rule 6860
Rubi steps
\begin {align*} \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{4+3 x^4+x^8} \, dx &=\int \left (\frac {\left (1+i \sqrt {7}\right ) \sqrt {2+x^4}}{3-i \sqrt {7}+2 x^4}+\frac {\left (1-i \sqrt {7}\right ) \sqrt {2+x^4}}{3+i \sqrt {7}+2 x^4}\right ) \, dx\\ &=\left (1-i \sqrt {7}\right ) \int \frac {\sqrt {2+x^4}}{3+i \sqrt {7}+2 x^4} \, dx+\left (1+i \sqrt {7}\right ) \int \frac {\sqrt {2+x^4}}{3-i \sqrt {7}+2 x^4} \, dx\\ &=\left (-3-i \sqrt {7}\right ) \int \frac {1}{\sqrt {2+x^4} \left (3+i \sqrt {7}+2 x^4\right )} \, dx+\frac {1}{2} \left (1-i \sqrt {7}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx+\left (-3+i \sqrt {7}\right ) \int \frac {1}{\sqrt {2+x^4} \left (3-i \sqrt {7}+2 x^4\right )} \, dx+\frac {1}{2} \left (1+i \sqrt {7}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx\\ &=\frac {\left (1-i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {2+x^4}}+\frac {\left (1+i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {1}{2} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (-3-i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (-3-i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (-3+i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (-3+i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx\\ &=\frac {\left (1-i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {2+x^4}}+\frac {\left (1+i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {\left (1-\frac {2}{\sqrt {-3-i \sqrt {7}}}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx}{2 \left (1+\frac {4 i}{3 i-\sqrt {7}}\right )}-\frac {\left (1+\frac {2}{\sqrt {-3-i \sqrt {7}}}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx}{2 \left (1+\frac {4 i}{3 i-\sqrt {7}}\right )}-\frac {\left (2-\sqrt {-3-i \sqrt {7}}\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (-3-i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx}{7+i \sqrt {7}}-\frac {\left (2+\sqrt {-3-i \sqrt {7}}\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (-3-i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx}{7+i \sqrt {7}}-\frac {\left (2-\sqrt {-3+i \sqrt {7}}\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (-3+i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx}{7-i \sqrt {7}}-\frac {\left (2+\sqrt {-3+i \sqrt {7}}\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (-3+i \sqrt {7}\right )}}\right ) \sqrt {2+x^4}} \, dx}{7-i \sqrt {7}}-\frac {\left (1-\frac {2}{\sqrt {-3+i \sqrt {7}}}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx}{2 \left (1+\frac {4 i}{3 i+\sqrt {7}}\right )}-\frac {\left (1+\frac {2}{\sqrt {-3+i \sqrt {7}}}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx}{2 \left (1+\frac {4 i}{3 i+\sqrt {7}}\right )}\\ &=-\frac {\sqrt [4]{-3-i \sqrt {7}} \tan ^{-1}\left (\frac {\sqrt {1-i \sqrt {7}} x}{\sqrt [4]{2 \left (-3-i \sqrt {7}\right )} \sqrt {2+x^4}}\right )}{2\ 2^{3/4} \sqrt {1-i \sqrt {7}}}-\frac {\sqrt [4]{-3+i \sqrt {7}} \tan ^{-1}\left (\frac {\sqrt {-1-i \sqrt {7}} x}{\sqrt [4]{2 \left (-3+i \sqrt {7}\right )} \sqrt {2+x^4}}\right )}{2\ 2^{3/4} \sqrt {-1-i \sqrt {7}}}-\frac {\sqrt [4]{-3-i \sqrt {7}} \tan ^{-1}\left (\frac {\sqrt {-1+i \sqrt {7}} x}{\sqrt [4]{2 \left (-3-i \sqrt {7}\right )} \sqrt {2+x^4}}\right )}{2\ 2^{3/4} \sqrt {-1+i \sqrt {7}}}-\frac {\sqrt [4]{-3+i \sqrt {7}} \tan ^{-1}\left (\frac {\sqrt {1+i \sqrt {7}} x}{\sqrt [4]{2 \left (-3+i \sqrt {7}\right )} \sqrt {2+x^4}}\right )}{2\ 2^{3/4} \sqrt {1+i \sqrt {7}}}+\frac {\left (1-i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {2+x^4}}+\frac {\left (1+i \sqrt {7}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {\left (1-\frac {2}{\sqrt {-3-i \sqrt {7}}}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1+\frac {4 i}{3 i-\sqrt {7}}\right ) \sqrt {2+x^4}}-\frac {\left (1+\frac {2}{\sqrt {-3-i \sqrt {7}}}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1+\frac {4 i}{3 i-\sqrt {7}}\right ) \sqrt {2+x^4}}-\frac {\left (1-\frac {2}{\sqrt {-3+i \sqrt {7}}}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1+\frac {4 i}{3 i+\sqrt {7}}\right ) \sqrt {2+x^4}}-\frac {\left (1+\frac {2}{\sqrt {-3+i \sqrt {7}}}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1+\frac {4 i}{3 i+\sqrt {7}}\right ) \sqrt {2+x^4}}-\frac {\left (2+\sqrt {-3-i \sqrt {7}}\right )^2 \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \Pi \left (-\frac {\left (2-\sqrt {-3-i \sqrt {7}}\right )^2}{8 \sqrt {-3-i \sqrt {7}}};2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7+i \sqrt {7}\right ) \sqrt {2+x^4}}-\frac {\left (2-\sqrt {-3-i \sqrt {7}}\right )^2 \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \Pi \left (\frac {\left (2+\sqrt {-3-i \sqrt {7}}\right )^2}{8 \sqrt {-3-i \sqrt {7}}};2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7+i \sqrt {7}\right ) \sqrt {2+x^4}}-\frac {\left (2+\sqrt {-3+i \sqrt {7}}\right )^2 \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \Pi \left (-\frac {\left (2-\sqrt {-3+i \sqrt {7}}\right )^2}{8 \sqrt {-3+i \sqrt {7}}};2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7-i \sqrt {7}\right ) \sqrt {2+x^4}}-\frac {\left (2-\sqrt {-3+i \sqrt {7}}\right )^2 \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \Pi \left (\frac {\left (2+\sqrt {-3+i \sqrt {7}}\right )^2}{8 \sqrt {-3+i \sqrt {7}}};2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{2} \left (7-i \sqrt {7}\right ) \sqrt {2+x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 33, normalized size = 0.18 \begin {gather*} -\frac {1}{2} \text {ArcTan}\left (\frac {x}{\sqrt {2+x^4}}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 0.52, size = 361, normalized size = 1.94
method | result | size |
elliptic | \(\frac {\left (\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+2}}{x}\right )}{2}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{4}+2}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(41\) |
trager | \(\frac {\ln \left (-\frac {-x^{4}+2 \sqrt {x^{4}+2}\, x -x^{2}-2}{x^{4}-x^{2}+2}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \sqrt {x^{4}+2}\, x +2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{2}+2}\right )}{4}\) | \(101\) |
default | \(\frac {\sqrt {2}\, \sqrt {4-2 i \sqrt {2}\, x^{2}}\, \sqrt {4+2 i \sqrt {2}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {i \sqrt {2}}}{2}, i\right )}{4 \sqrt {i \sqrt {2}}\, \sqrt {x^{4}+2}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {4 \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}+1\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+2}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {2 \,2^{\frac {1}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {2-i \sqrt {2}\, x^{2}}\, \sqrt {2+i \sqrt {2}\, x^{2}}\, \EllipticPi \left (\frac {x \sqrt {2}\, \sqrt {i \sqrt {2}}}{2}, \frac {i \sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}-\frac {i \sqrt {2}}{2}, \frac {\sqrt {-\frac {i \sqrt {2}}{2}}\, \sqrt {2}}{\sqrt {i \sqrt {2}}}\right )}{\sqrt {i}\, \sqrt {x^{4}+2}}\right )\right )}{32}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{2}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {4 \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-1\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+2}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {2 \,2^{\frac {1}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {2-i \sqrt {2}\, x^{2}}\, \sqrt {2+i \sqrt {2}\, x^{2}}\, \EllipticPi \left (\frac {x \sqrt {2}\, \sqrt {i \sqrt {2}}}{2}, \frac {i \sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {i \sqrt {2}}{2}, \frac {\sqrt {-\frac {i \sqrt {2}}{2}}\, \sqrt {2}}{\sqrt {i \sqrt {2}}}\right )}{\sqrt {i}\, \sqrt {x^{4}+2}}\right )\right )}{32}\) | \(361\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 0.42, size = 60, normalized size = 0.32 \begin {gather*} -\frac {1}{4} \, \arctan \left (\frac {2 \, \sqrt {x^{4} + 2} x}{x^{4} - x^{2} + 2}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} + x^{2} - 2 \, \sqrt {x^{4} + 2} x + 2}{x^{4} - x^{2} + 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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^4-2\right )\,\sqrt {x^4+2}}{x^8+3\,x^4+4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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