Optimal. Leaf size=101 \[ \frac {2 \sqrt [4]{1+x^4}}{x}-\frac {1}{8} \text {RootSum}\left [2-4 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-4 \log (x)+4 \log \left (\sqrt [4]{1+x^4}-x \text {$\#$1}\right )+7 \log (x) \text {$\#$1}^4-7 \log \left (\sqrt [4]{1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}^3+\text {$\#$1}^7}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(425\) vs. \(2(101)=202\).
time = 0.94, antiderivative size = 425, normalized size of antiderivative = 4.21, number of steps
used = 40, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6860, 283,
338, 304, 209, 212, 1542, 524, 1532, 508} \begin {gather*} -\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )+\frac {1}{4} \sqrt [4]{10+7 \sqrt {2}} \text {ArcTan}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{4} \sqrt [4]{2-\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{4} \sqrt [4]{2+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{4} \sqrt [4]{10-7 \sqrt {2}} \text {ArcTan}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )+\frac {2 \sqrt [4]{x^4+1}}{x}-\frac {1}{4} \sqrt [4]{10+7 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{4} \sqrt [4]{2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{4} \sqrt [4]{2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{4} \sqrt [4]{10-7 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{x^4+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 283
Rule 304
Rule 338
Rule 508
Rule 524
Rule 1532
Rule 1542
Rule 6860
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+x^4} \left (2+x^4\right )}{x^2 \left (-1+2 x^4+x^8\right )} \, dx &=\int \left (-\frac {2 \sqrt [4]{1+x^4}}{x^2}+\frac {x^2 \sqrt [4]{1+x^4} \left (5+2 x^4\right )}{-1+2 x^4+x^8}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt [4]{1+x^4}}{x^2} \, dx\right )+\int \frac {x^2 \sqrt [4]{1+x^4} \left (5+2 x^4\right )}{-1+2 x^4+x^8} \, dx\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-2 \int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx+\int \left (\frac {5 x^2 \sqrt [4]{1+x^4}}{-1+2 x^4+x^8}+\frac {2 x^6 \sqrt [4]{1+x^4}}{-1+2 x^4+x^8}\right ) \, dx\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}+2 \int \frac {x^6 \sqrt [4]{1+x^4}}{-1+2 x^4+x^8} \, dx-2 \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+5 \int \frac {x^2 \sqrt [4]{1+x^4}}{-1+2 x^4+x^8} \, dx\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}+2 \int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx-2 \int \frac {x^2 \left (-1+x^4\right )}{\left (1+x^4\right )^{3/4} \left (-1+2 x^4+x^8\right )} \, dx+5 \int \left (-\frac {x^2 \sqrt [4]{1+x^4}}{\sqrt {2} \left (-2+2 \sqrt {2}-2 x^4\right )}-\frac {x^2 \sqrt [4]{1+x^4}}{\sqrt {2} \left (2+2 \sqrt {2}+2 x^4\right )}\right ) \, dx-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}+\tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-2 \int \left (-\frac {x^2}{\left (1+x^4\right )^{3/4} \left (-1+2 x^4+x^8\right )}+\frac {x^6}{\left (1+x^4\right )^{3/4} \left (-1+2 x^4+x^8\right )}\right ) \, dx+2 \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {5 \int \frac {x^2 \sqrt [4]{1+x^4}}{-2+2 \sqrt {2}-2 x^4} \, dx}{\sqrt {2}}-\frac {5 \int \frac {x^2 \sqrt [4]{1+x^4}}{2+2 \sqrt {2}+2 x^4} \, dx}{\sqrt {2}}\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )+\tan ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+2 \int \frac {x^2}{\left (1+x^4\right )^{3/4} \left (-1+2 x^4+x^8\right )} \, dx-2 \int \frac {x^6}{\left (1+x^4\right )^{3/4} \left (-1+2 x^4+x^8\right )} \, dx+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )+2 \int \left (-\frac {x^2}{\sqrt {2} \left (-2+2 \sqrt {2}-2 x^4\right ) \left (1+x^4\right )^{3/4}}-\frac {x^2}{\sqrt {2} \left (1+x^4\right )^{3/4} \left (2+2 \sqrt {2}+2 x^4\right )}\right ) \, dx-2 \int \left (-\frac {\left (-2+2 \sqrt {2}\right ) x^2}{2 \sqrt {2} \left (-2+2 \sqrt {2}-2 x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {\left (2+2 \sqrt {2}\right ) x^2}{2 \sqrt {2} \left (1+x^4\right )^{3/4} \left (2+2 \sqrt {2}+2 x^4\right )}\right ) \, dx\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )-\sqrt {2} \int \frac {x^2}{\left (-2+2 \sqrt {2}-2 x^4\right ) \left (1+x^4\right )^{3/4}} \, dx-\sqrt {2} \int \frac {x^2}{\left (1+x^4\right )^{3/4} \left (2+2 \sqrt {2}+2 x^4\right )} \, dx-\left (-2+\sqrt {2}\right ) \int \frac {x^2}{\left (-2+2 \sqrt {2}-2 x^4\right ) \left (1+x^4\right )^{3/4}} \, dx-\left (2+\sqrt {2}\right ) \int \frac {x^2}{\left (1+x^4\right )^{3/4} \left (2+2 \sqrt {2}+2 x^4\right )} \, dx\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )-\sqrt {2} \text {Subst}\left (\int \frac {x^2}{-2+2 \sqrt {2}-2 \sqrt {2} x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\sqrt {2} \text {Subst}\left (\int \frac {x^2}{2+2 \sqrt {2}-2 \sqrt {2} x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\left (-2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {x^2}{-2+2 \sqrt {2}-2 \sqrt {2} x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {x^2}{2+2 \sqrt {2}-2 \sqrt {2} x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {2}}-\frac {1}{4} \left (2-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{4} \left (-2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{4} \left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{4} \left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )\\ &=\frac {2 \sqrt [4]{1+x^4}}{x}-\frac {5}{12} \left (2-\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};-x^4,\left (1-\sqrt {2}\right ) x^4\right )-\frac {5}{12} \left (2+\sqrt {2}\right ) x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\left (1+\sqrt {2}\right ) x^4,-x^4\right )+\frac {1}{4} \sqrt [4]{2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )+\frac {\left (2+\sqrt {2}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}}-\frac {\left (2-\sqrt {2}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}}+\frac {1}{4} \sqrt [4]{2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{4} \sqrt [4]{2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )-\frac {\left (2+\sqrt {2}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2-\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}}+\frac {1}{4} \sqrt [4]{10-7 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{4} \sqrt [4]{2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{2+\sqrt {2}} x}{\sqrt [4]{1+x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 101, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{1+x^4}}{x}-\frac {1}{8} \text {RootSum}\left [2-4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-4 \log (x)+4 \log \left (\sqrt [4]{1+x^4}-x \text {$\#$1}\right )+7 \log (x) \text {$\#$1}^4-7 \log \left (\sqrt [4]{1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-2 \text {$\#$1}^3+\text {$\#$1}^7}\&\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 = 33.87, size = 5218, normalized size = 51.66 \[\text {output too large to display}\]
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 = 5.73, size = 1214, normalized size = 12.02 \begin {gather*} \frac {4 \, \sqrt {2} x {\left (1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} \arctan \left (-\frac {{\left (\sqrt {2} {\left (69 \, x^{8} + 18 \, x^{4} + 2 \, {\left (1451 \, x^{6} - 601 \, x^{2} - \sqrt {2} {\left (1026 \, x^{6} - 425 \, x^{2}\right )}\right )} \sqrt {x^{4} + 1} \sqrt {1393 \, \sqrt {2} + 1970} - \sqrt {2} {\left (47 \, x^{8} + 8 \, x^{4} - 13\right )} - 17\right )} \sqrt {-{\left (2671 \, \sqrt {2} - 3778\right )} \sqrt {1393 \, \sqrt {2} + 1970}} - 196 \, {\left (7 \, x^{5} - \sqrt {2} {\left (5 \, x^{5} - 2 \, x\right )} - 3 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} - 196 \, {\left (239 \, x^{7} - 99 \, x^{3} - \sqrt {2} {\left (169 \, x^{7} - 70 \, x^{3}\right )}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {1393 \, \sqrt {2} + 1970}\right )} {\left (1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}}}{98 \, {\left (x^{8} + 2 \, x^{4} - 1\right )}}\right ) + 4 \, \sqrt {2} x {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (69 \, x^{8} + 18 \, x^{4} + 2 \, {\left (1451 \, x^{6} - 601 \, x^{2} + \sqrt {2} {\left (1026 \, x^{6} - 425 \, x^{2}\right )}\right )} \sqrt {x^{4} + 1} \sqrt {-1393 \, \sqrt {2} + 1970} + \sqrt {2} {\left (47 \, x^{8} + 8 \, x^{4} - 13\right )} - 17\right )} \sqrt {{\left (2671 \, \sqrt {2} + 3778\right )} \sqrt {-1393 \, \sqrt {2} + 1970}} {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} - 196 \, {\left ({\left (7 \, x^{5} + \sqrt {2} {\left (5 \, x^{5} - 2 \, x\right )} - 3 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + {\left (239 \, x^{7} - 99 \, x^{3} + \sqrt {2} {\left (169 \, x^{7} - 70 \, x^{3}\right )}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {-1393 \, \sqrt {2} + 1970}\right )} {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}}}{98 \, {\left (x^{8} + 2 \, x^{4} - 1\right )}}\right ) + \sqrt {2} x {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} + \sqrt {2} {\left (6 \, x^{5} - 5 \, x\right )} - x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 4 \, {\left (331 \, x^{7} - 137 \, x^{3} + \sqrt {2} {\left (234 \, x^{7} - 97 \, x^{3}\right )}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {-1393 \, \sqrt {2} + 1970} + {\left (4 \, {\left (57 \, x^{6} - 23 \, x^{2} + \sqrt {2} {\left (40 \, x^{6} - 17 \, x^{2}\right )}\right )} \sqrt {x^{4} + 1} + {\left (10444 \, x^{8} + 2260 \, x^{4} + \sqrt {2} {\left (7385 \, x^{8} + 1598 \, x^{4} - 1929\right )} - 2728\right )} \sqrt {-1393 \, \sqrt {2} + 1970}\right )} {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}}}{x^{8} + 2 \, x^{4} - 1}\right ) - \sqrt {2} x {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} + \sqrt {2} {\left (6 \, x^{5} - 5 \, x\right )} - x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 4 \, {\left (331 \, x^{7} - 137 \, x^{3} + \sqrt {2} {\left (234 \, x^{7} - 97 \, x^{3}\right )}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {-1393 \, \sqrt {2} + 1970} - {\left (4 \, {\left (57 \, x^{6} - 23 \, x^{2} + \sqrt {2} {\left (40 \, x^{6} - 17 \, x^{2}\right )}\right )} \sqrt {x^{4} + 1} + {\left (10444 \, x^{8} + 2260 \, x^{4} + \sqrt {2} {\left (7385 \, x^{8} + 1598 \, x^{4} - 1929\right )} - 2728\right )} \sqrt {-1393 \, \sqrt {2} + 1970}\right )} {\left (-1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}}}{x^{8} + 2 \, x^{4} - 1}\right ) - \sqrt {2} x {\left (1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} - \sqrt {2} {\left (6 \, x^{5} - 5 \, x\right )} - x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 4 \, {\left (331 \, x^{7} - 137 \, x^{3} - \sqrt {2} {\left (234 \, x^{7} - 97 \, x^{3}\right )}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {1393 \, \sqrt {2} + 1970} + {\left (4 \, {\left (57 \, x^{6} - 23 \, x^{2} - \sqrt {2} {\left (40 \, x^{6} - 17 \, x^{2}\right )}\right )} \sqrt {x^{4} + 1} + {\left (10444 \, x^{8} + 2260 \, x^{4} - \sqrt {2} {\left (7385 \, x^{8} + 1598 \, x^{4} - 1929\right )} - 2728\right )} \sqrt {1393 \, \sqrt {2} + 1970}\right )} {\left (1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}}}{x^{8} + 2 \, x^{4} - 1}\right ) + \sqrt {2} x {\left (1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} - \sqrt {2} {\left (6 \, x^{5} - 5 \, x\right )} - x\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 4 \, {\left (331 \, x^{7} - 137 \, x^{3} - \sqrt {2} {\left (234 \, x^{7} - 97 \, x^{3}\right )}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} \sqrt {1393 \, \sqrt {2} + 1970} - {\left (4 \, {\left (57 \, x^{6} - 23 \, x^{2} - \sqrt {2} {\left (40 \, x^{6} - 17 \, x^{2}\right )}\right )} \sqrt {x^{4} + 1} + {\left (10444 \, x^{8} + 2260 \, x^{4} - \sqrt {2} {\left (7385 \, x^{8} + 1598 \, x^{4} - 1929\right )} - 2728\right )} \sqrt {1393 \, \sqrt {2} + 1970}\right )} {\left (1393 \, \sqrt {2} + 1970\right )}^{\frac {1}{4}}}{x^{8} + 2 \, x^{4} - 1}\right ) + 64 \, {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{32 \, x} \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+1\right )}^{1/4}\,\left (x^4+2\right )}{x^2\,\left (x^8+2\,x^4-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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