Optimal. Leaf size=49 \[ \frac {1}{8} \text {RootSum}\left [1-4 \text {$\#$1}^4+2 \text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(277\) vs. \(2(49)=98\).
time = 0.12, antiderivative size = 277, normalized size of antiderivative = 5.65, number of steps
used = 11, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2081, 1284,
1443, 385, 218, 212, 209} \begin {gather*} -\frac {\sqrt [4]{2-\sqrt {2}} \sqrt {x} \sqrt [4]{x^2+1} \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{4 \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{2+\sqrt {2}} \sqrt {x} \sqrt [4]{x^2+1} \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{4 \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{2-\sqrt {2}} \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{4 \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{2+\sqrt {2}} \sqrt {x} \sqrt [4]{x^2+1} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{4 \sqrt [4]{x^4+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 385
Rule 1284
Rule 1443
Rule 2081
Rubi steps
\begin {align*} \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{1+x^2} \left (-2+x^4\right )} \, dx}{\sqrt [4]{x^2+x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (-2+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {2}-x^4\right ) \sqrt [4]{1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (\sqrt {2}+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}-\left (-1+\sqrt {2}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {2} \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}-\left (1+\sqrt {2}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {2} \sqrt [4]{x^2+x^4}}\\ &=-\frac {\left (\sqrt {2-\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {2-\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {2+\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {2+\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}\\ &=-\frac {\sqrt [4]{2-\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}-\frac {\sqrt [4]{2+\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}-\frac {\sqrt [4]{2-\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}-\frac {\sqrt [4]{2+\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 80, normalized size = 1.63 \begin {gather*} \frac {\sqrt {x} \sqrt [4]{1+x^2} \text {RootSum}\left [1-4 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 \sqrt [4]{x^2+x^4}} \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 = 44.04, size = 3152, normalized size = 64.33
method | result | size |
trager | \(\text {Expression too large to display}\) | \(3152\) |
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 = 3.13, size = 1118, normalized size = 22.82 \begin {gather*} -\frac {1}{4} \, {\left (\sqrt {2} + 2\right )}^{\frac {1}{4}} \arctan \left (-\frac {{\left (196 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {\sqrt {2} + 2} + \sqrt {2} {\left (2 \, \sqrt {x^{4} + x^{2}} {\left (10 \, x^{3} + \sqrt {2} {\left (x^{3} - 10 \, x\right )} - 2 \, x\right )} + {\left (19 \, x^{5} + 16 \, x^{3} - \sqrt {2} {\left (3 \, x^{5} + 18 \, x^{3} + 10 \, x\right )} - 2 \, x\right )} \sqrt {\sqrt {2} + 2}\right )} \sqrt {-{\left (132 \, \sqrt {2} - 193\right )} \sqrt {\sqrt {2} + 2}} + 196 \, {\left (x^{4} + 2 \, x^{2} - \sqrt {2} {\left (x^{4} + x^{2}\right )}\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}\right )} {\left (\sqrt {2} + 2\right )}^{\frac {1}{4}}}{98 \, {\left (x^{5} - 2 \, x\right )}}\right ) - \frac {1}{4} \, {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (2 \, \sqrt {x^{4} + x^{2}} {\left (10 \, x^{3} - \sqrt {2} {\left (x^{3} - 10 \, x\right )} - 2 \, x\right )} + {\left (19 \, x^{5} + 16 \, x^{3} + \sqrt {2} {\left (3 \, x^{5} + 18 \, x^{3} + 10 \, x\right )} - 2 \, x\right )} \sqrt {-\sqrt {2} + 2}\right )} \sqrt {{\left (132 \, \sqrt {2} + 193\right )} \sqrt {-\sqrt {2} + 2}} {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}} + 196 \, {\left ({\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {-\sqrt {2} + 2} + {\left (x^{4} + 2 \, x^{2} + \sqrt {2} {\left (x^{4} + x^{2}\right )}\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}\right )} {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}}}{98 \, {\left (x^{5} - 2 \, x\right )}}\right ) - \frac {1}{16} \, {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (11 \, x^{2} + \sqrt {2} {\left (6 \, x^{2} + 11\right )} + 12\right )} + 2 \, {\left (34 \, x^{4} + 46 \, x^{2} + \sqrt {2} {\left (23 \, x^{4} + 34 \, x^{2}\right )}\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} + {\left (56 \, x^{5} + 92 \, x^{3} + 2 \, \sqrt {x^{4} + x^{2}} {\left (34 \, x^{3} + \sqrt {2} {\left (23 \, x^{3} + 34 \, x\right )} + 46 \, x\right )} \sqrt {-\sqrt {2} + 2} + \sqrt {2} {\left (35 \, x^{5} + 68 \, x^{3} + 22 \, x\right )} + 24 \, x\right )} {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}}}{x^{5} - 2 \, x}\right ) + \frac {1}{16} \, {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (11 \, x^{2} + \sqrt {2} {\left (6 \, x^{2} + 11\right )} + 12\right )} + 2 \, {\left (34 \, x^{4} + 46 \, x^{2} + \sqrt {2} {\left (23 \, x^{4} + 34 \, x^{2}\right )}\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} - {\left (56 \, x^{5} + 92 \, x^{3} + 2 \, \sqrt {x^{4} + x^{2}} {\left (34 \, x^{3} + \sqrt {2} {\left (23 \, x^{3} + 34 \, x\right )} + 46 \, x\right )} \sqrt {-\sqrt {2} + 2} + \sqrt {2} {\left (35 \, x^{5} + 68 \, x^{3} + 22 \, x\right )} + 24 \, x\right )} {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}}}{x^{5} - 2 \, x}\right ) - \frac {1}{16} \, {\left (\sqrt {2} + 2\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (11 \, x^{2} - \sqrt {2} {\left (6 \, x^{2} + 11\right )} + 12\right )} + 2 \, {\left (34 \, x^{4} + 46 \, x^{2} - \sqrt {2} {\left (23 \, x^{4} + 34 \, x^{2}\right )}\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + {\left (56 \, x^{5} + 92 \, x^{3} + 2 \, \sqrt {x^{4} + x^{2}} {\left (34 \, x^{3} - \sqrt {2} {\left (23 \, x^{3} + 34 \, x\right )} + 46 \, x\right )} \sqrt {\sqrt {2} + 2} - \sqrt {2} {\left (35 \, x^{5} + 68 \, x^{3} + 22 \, x\right )} + 24 \, x\right )} {\left (\sqrt {2} + 2\right )}^{\frac {1}{4}}}{x^{5} - 2 \, x}\right ) + \frac {1}{16} \, {\left (\sqrt {2} + 2\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (11 \, x^{2} - \sqrt {2} {\left (6 \, x^{2} + 11\right )} + 12\right )} + 2 \, {\left (34 \, x^{4} + 46 \, x^{2} - \sqrt {2} {\left (23 \, x^{4} + 34 \, x^{2}\right )}\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - {\left (56 \, x^{5} + 92 \, x^{3} + 2 \, \sqrt {x^{4} + x^{2}} {\left (34 \, x^{3} - \sqrt {2} {\left (23 \, x^{3} + 34 \, x\right )} + 46 \, x\right )} \sqrt {\sqrt {2} + 2} - \sqrt {2} {\left (35 \, x^{5} + 68 \, x^{3} + 22 \, x\right )} + 24 \, x\right )} {\left (\sqrt {2} + 2\right )}^{\frac {1}{4}}}{x^{5} - 2 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} - 2\right )}\, dx \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 = -0.02 \begin {gather*} \int \frac {1}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^4-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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