Optimal. Leaf size=248 \[ 2 \sqrt [4]{2} \text {RootSum}\left [\text {$\#$1}^8+12 \sqrt {2} \text {$\#$1}^6-4 \text {$\#$1}^6-24 \sqrt {2} \text {$\#$1}^4+70 \text {$\#$1}^4+12 \sqrt {2} \text {$\#$1}^2-4 \text {$\#$1}^2+1\& ,\frac {\text {$\#$1}^3 \left (-\log \left (-x^2+2^{3/4} x-\sqrt {2}\right )\right )+\text {$\#$1}^3 \log \left (\text {$\#$1} x^2-2^{3/4} \text {$\#$1} x+\sqrt {2} \text {$\#$1}+\sqrt {x^4+2}\right )-\text {$\#$1} \log \left (-x^2+2^{3/4} x-\sqrt {2}\right )+\text {$\#$1} \log \left (\text {$\#$1} x^2-2^{3/4} \text {$\#$1} x+\sqrt {2} \text {$\#$1}+\sqrt {x^4+2}\right )}{\text {$\#$1}^6+9 \sqrt {2} \text {$\#$1}^4-3 \text {$\#$1}^4-12 \sqrt {2} \text {$\#$1}^2+35 \text {$\#$1}^2+3 \sqrt {2}-1}\& \right ] \]
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Rubi [C] time = 1.84, antiderivative size = 905, normalized size of antiderivative = 3.65, number of steps used = 34, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {6728, 1209, 1198, 220, 1196, 1217, 1707} \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+2}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+2}}\right )-\frac {\left (i-\sqrt {7}\right ) \left (1+2 \sqrt {2}+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 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (7 i+\sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (1+2 \sqrt {2}-i \sqrt {7}\right ) \left (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 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (7 i-\sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (1-2 \sqrt {2}+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 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left (1-2 \sqrt {2}-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 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {x^4+2}}+\frac {i \left ((1+i)+\sqrt {2}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left ((-1+i)+\sqrt {2}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left ((-1-i)+\sqrt {2}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left ((1+i)+i \sqrt {2}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {x^4+2}}-\frac {\left (1+2 \sqrt {2}\right ) \left (3 i-\sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \Pi \left (\frac {1}{8} \left (4-\sqrt {2}\right );2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (7 i+3 \sqrt {7}\right ) \sqrt {x^4+2}}-\frac {\left (1+2 \sqrt {2}\right ) \left (3 i+\sqrt {7}\right ) \left (x^2+\sqrt {2}\right ) \sqrt {\frac {x^4+2}{\left (x^2+\sqrt {2}\right )^2}} \Pi \left (\frac {1}{8} \left (4-\sqrt {2}\right );2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (7 i-3 \sqrt {7}\right ) \sqrt {x^4+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 220
Rule 1196
Rule 1198
Rule 1209
Rule 1217
Rule 1707
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-2+x^4\right ) \sqrt {2+x^4}}{\left (2+x^2+x^4\right ) \left (2+2 x^2+x^4\right )} \, dx &=\int \left (\frac {\left (1+2 x^2\right ) \sqrt {2+x^4}}{2+x^2+x^4}-\frac {2 \left (1+x^2\right ) \sqrt {2+x^4}}{2+2 x^2+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {\left (1+x^2\right ) \sqrt {2+x^4}}{2+2 x^2+x^4} \, dx\right )+\int \frac {\left (1+2 x^2\right ) \sqrt {2+x^4}}{2+x^2+x^4} \, dx\\ &=-\left (2 \int \left (\frac {\sqrt {2+x^4}}{(2-2 i)+2 x^2}+\frac {\sqrt {2+x^4}}{(2+2 i)+2 x^2}\right ) \, dx\right )+\int \left (\frac {2 \sqrt {2+x^4}}{1-i \sqrt {7}+2 x^2}+\frac {2 \sqrt {2+x^4}}{1+i \sqrt {7}+2 x^2}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {2+x^4}}{(2-2 i)+2 x^2} \, dx\right )-2 \int \frac {\sqrt {2+x^4}}{(2+2 i)+2 x^2} \, dx+2 \int \frac {\sqrt {2+x^4}}{1-i \sqrt {7}+2 x^2} \, dx+2 \int \frac {\sqrt {2+x^4}}{1+i \sqrt {7}+2 x^2} \, dx\\ &=\frac {1}{2} \int \frac {(2-2 i)-2 x^2}{\sqrt {2+x^4}} \, dx+\frac {1}{2} \int \frac {(2+2 i)-2 x^2}{\sqrt {2+x^4}} \, dx-\frac {1}{2} \int \frac {1-i \sqrt {7}-2 x^2}{\sqrt {2+x^4}} \, dx-\frac {1}{2} \int \frac {1+i \sqrt {7}-2 x^2}{\sqrt {2+x^4}} \, dx-(4-4 i) \int \frac {1}{\left ((2-2 i)+2 x^2\right ) \sqrt {2+x^4}} \, dx-(4+4 i) \int \frac {1}{\left ((2+2 i)+2 x^2\right ) \sqrt {2+x^4}} \, dx+\left (1-i \sqrt {7}\right ) \int \frac {1}{\left (1-i \sqrt {7}+2 x^2\right ) \sqrt {2+x^4}} \, dx+\left (1+i \sqrt {7}\right ) \int \frac {1}{\left (1+i \sqrt {7}+2 x^2\right ) \sqrt {2+x^4}} \, dx\\ &=\left ((1-i)-\sqrt {2}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx+\left ((1+i)-\sqrt {2}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx-\left ((1+i)+i \sqrt {2}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx-\left ((-2-2 i) \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left ((2-2 i)+2 x^2\right ) \sqrt {2+x^4}} \, dx+\left (i \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx-\left (2 i \left (2+(1+i) \sqrt {2}\right )\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left ((2+2 i)+2 x^2\right ) \sqrt {2+x^4}} \, dx-\frac {1}{2} \left (1-2 \sqrt {2}-i \sqrt {7}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx-\frac {1}{2} \left (1-2 \sqrt {2}+i \sqrt {7}\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx+\frac {\left (\left (-1+i \sqrt {7}\right ) \left (i+2 i \sqrt {2}+\sqrt {7}\right )\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx}{2 \left (7 i-\sqrt {7}\right )}+\frac {\left (4 \left (-1+i \sqrt {7}\right ) \left (2+\frac {1-i \sqrt {7}}{\sqrt {2}}\right )\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left (1-i \sqrt {7}+2 x^2\right ) \sqrt {2+x^4}} \, dx}{-8+\left (1-i \sqrt {7}\right )^2}+\frac {\left (\left (1+i \sqrt {7}\right ) \left (1+2 \sqrt {2}+i \sqrt {7}\right )\right ) \int \frac {1}{\sqrt {2+x^4}} \, dx}{-8+\left (1+i \sqrt {7}\right )^2}+\frac {\left (4 \left (-1-i \sqrt {7}\right ) \left (2+\frac {1+i \sqrt {7}}{\sqrt {2}}\right )\right ) \int \frac {1+\frac {x^2}{\sqrt {2}}}{\left (1+i \sqrt {7}+2 x^2\right ) \sqrt {2+x^4}} \, dx}{-8+\left (1+i \sqrt {7}\right )^2}\\ &=\tan ^{-1}\left (\frac {x}{\sqrt {2+x^4}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {2+x^4}}\right )-\frac {\left ((1+i)+i \sqrt {2}\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 )}{2 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {\left ((-1-i)+\sqrt {2}\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 )}{2 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {\left ((-1+i)+\sqrt {2}\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 )}{2 \sqrt [4]{2} \sqrt {2+x^4}}+\frac {i \left ((1+i)+\sqrt {2}\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 )}{2 \sqrt [4]{2} \sqrt {2+x^4}}-\frac {\left (1-2 \sqrt {2}-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 (1-2 \sqrt {2}+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+2 \sqrt {2}-i \sqrt {7}\right ) \left (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 (7 i-\sqrt {7}\right ) \sqrt {2+x^4}}-\frac {\left (i-\sqrt {7}\right ) \left (1+2 \sqrt {2}+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 (7 i+\sqrt {7}\right ) \sqrt {2+x^4}}-\frac {i \left (1+\sqrt {2}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2-\sqrt {2}\right );2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \sqrt {2+x^4}}+\frac {i \left (2+\sqrt {2}\right ) \left (\sqrt {2}+x^2\right ) \sqrt {\frac {2+x^4}{\left (\sqrt {2}+x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2-\sqrt {2}\right );2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2\ 2^{3/4} \sqrt {2+x^4}}+\frac {\left (1+2 \sqrt {2}-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 {1}{8} \left (4-\sqrt {2}\right );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 (1+2 \sqrt {2}+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 {1}{8} \left (4-\sqrt {2}\right );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 [C] time = 0.89, size = 148, normalized size = 0.60 \begin {gather*} -\sqrt [4]{-\frac {1}{2}} \left (F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {1}{2}} x\right )\right |-1\right )-2 \Pi \left (-\frac {1+i}{\sqrt {2}};\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {1}{2}} x\right )\right |-1\right )-2 \Pi \left (\frac {1-i}{\sqrt {2}};\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {1}{2}} x\right )\right |-1\right )+\Pi \left (-\frac {2 \sqrt {2}}{-i+\sqrt {7}};\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {1}{2}} x\right )\right |-1\right )+\Pi \left (\frac {2 \sqrt {2}}{i+\sqrt {7}};\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {1}{2}} x\right )\right |-1\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.98, size = 37, normalized size = 0.15 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt {2+x^4}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 57, normalized size = 0.23 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{4} + 2} x}{x^{4} - 2 \, x^{2} + 2}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{4} + 2} x}{x^{4} - x^{2} + 2}\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 {x^{4} + 2} {\left (x^{4} - 2\right )}}{{\left (x^{4} + 2 \, x^{2} + 2\right )} {\left (x^{4} + x^{2} + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.74, size = 42, normalized size = 0.17
method | result | size |
elliptic | \(\frac {\left (2 \arctan \left (\frac {\sqrt {x^{4}+2}\, \sqrt {2}}{2 x}\right )-\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+2}}{x}\right )\right ) \sqrt {2}}{2}\) | \(42\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{4}+2}\, x +2 \RootOf \left (\textit {\_Z}^{2}+2\right )}{x^{4}+2 x^{2}+2}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \sqrt {x^{4}+2}\, x +2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{2}+2}\right )}{2}\) | \(125\) |
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 {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {2 \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-2\right )}{\sqrt {-2 \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+2}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {2^{\frac {3}{4}} \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-2 \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}+i \sqrt {2}, \frac {\sqrt {-\frac {i \sqrt {2}}{2}}\, \sqrt {2}}{\sqrt {i \sqrt {2}}}\right )}{\sqrt {i}\, \sqrt {x^{4}+2}}\right )\right )}{8}-\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 )}{16}\) | \(370\) |
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} + 2} {\left (x^{4} - 2\right )}}{{\left (x^{4} + 2 \, x^{2} + 2\right )} {\left (x^{4} + x^{2} + 2\right )}}\,{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 = -0.00 \begin {gather*} \int \frac {\left (x^4-2\right )\,\sqrt {x^4+2}}{\left (x^4+x^2+2\right )\,\left (x^4+2\,x^2+2\right )} \,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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