Optimal. Leaf size=86 \[ -\frac {1}{16} \text {RootSum}\left [1-4 \text {$\#$1}^4+2 \text {$\#$1}^8\& ,\frac {-3 \log (x)+3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\& \right ] \]
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Rubi [A]
time = 0.19, antiderivative size = 171, normalized size of antiderivative = 1.99, number of steps
used = 10, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6857, 385,
218, 212, 209} \begin {gather*} -\frac {\sqrt [4]{58-41 \sqrt {2}} \text {ArcTan}\left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}-\frac {1}{8} \left (2+\sqrt {2}\right )^{5/4} \text {ArcTan}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{58-41 \sqrt {2}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}-\frac {1}{8} \left (2+\sqrt {2}\right )^{5/4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^4-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 385
Rule 6857
Rubi steps
\begin {align*} \int \frac {2+x^4}{\sqrt [4]{-1+x^4} \left (-2+x^8\right )} \, dx &=\int \left (-\frac {2+\sqrt {2}}{2 \sqrt {2} \left (\sqrt {2}-x^4\right ) \sqrt [4]{-1+x^4}}+\frac {-2+\sqrt {2}}{2 \sqrt {2} \sqrt [4]{-1+x^4} \left (\sqrt {2}+x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (\sqrt {2}+x^4\right )} \, dx-\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}-x^4\right ) \sqrt [4]{-1+x^4}} \, dx\\ &=\frac {1}{2} \left (1-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}-\left (1+\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}-\left (-1+\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {1}{4} \left (\left (1-\sqrt {2}\right ) \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{4} \left (\left (1-\sqrt {2}\right ) \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{4} \left (\left (1+\sqrt {2}\right ) \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{4} \left (\left (1+\sqrt {2}\right ) \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=-\frac {\sqrt [4]{58-41 \sqrt {2}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\sqrt [4]{58+41 \sqrt {2}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\sqrt [4]{58-41 \sqrt {2}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\sqrt [4]{58+41 \sqrt {2}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 86, normalized size = 1.00 \begin {gather*} -\frac {1}{16} \text {RootSum}\left [1-4 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-3 \log (x)+3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\&\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 = 13.53, size = 2578, normalized size = 29.98
method | result | size |
trager | \(\text {Expression too large to display}\) | \(2578\) |
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 = 8.29, size = 1163, normalized size = 13.52 \begin {gather*} \frac {1}{8} \, \sqrt {2} {\left (41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} \arctan \left (-\frac {{\left (196 \, {\left (17 \, x^{5} - \sqrt {2} {\left (12 \, x^{5} - 17 \, x\right )} - 24 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} \sqrt {41 \, \sqrt {2} + 58} + \sqrt {2} {\left (2 \, {\left (8 \, x^{6} - 18 \, x^{2} - \sqrt {2} {\left (9 \, x^{6} - 8 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} + {\left (163 \, x^{8} - 292 \, x^{4} - 2 \, \sqrt {2} {\left (58 \, x^{8} - 103 \, x^{4} + 30\right )} + 86\right )} \sqrt {41 \, \sqrt {2} + 58}\right )} \sqrt {-{\left (782 \, \sqrt {2} - 1107\right )} \sqrt {41 \, \sqrt {2} + 58}} + 196 \, {\left (3 \, x^{7} - 4 \, x^{3} - \sqrt {2} {\left (2 \, x^{7} - 3 \, x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )} {\left (41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}}}{98 \, {\left (x^{8} - 2\right )}}\right ) + \frac {1}{8} \, \sqrt {2} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, {\left (8 \, x^{6} - 18 \, x^{2} + \sqrt {2} {\left (9 \, x^{6} - 8 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} + {\left (163 \, x^{8} - 292 \, x^{4} + 2 \, \sqrt {2} {\left (58 \, x^{8} - 103 \, x^{4} + 30\right )} + 86\right )} \sqrt {-41 \, \sqrt {2} + 58}\right )} \sqrt {{\left (782 \, \sqrt {2} + 1107\right )} \sqrt {-41 \, \sqrt {2} + 58}} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} + 196 \, {\left ({\left (17 \, x^{5} + \sqrt {2} {\left (12 \, x^{5} - 17 \, x\right )} - 24 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} \sqrt {-41 \, \sqrt {2} + 58} + {\left (3 \, x^{7} - 4 \, x^{3} + \sqrt {2} {\left (2 \, x^{7} - 3 \, x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}}}{98 \, {\left (x^{8} - 2\right )}}\right ) - \frac {1}{32} \, \sqrt {2} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} + \sqrt {2} {\left (6 \, x^{5} - 11 \, x\right )} - 12 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (194 \, x^{7} - 274 \, x^{3} + \sqrt {2} {\left (137 \, x^{7} - 194 \, x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} \sqrt {-41 \, \sqrt {2} + 58} + {\left (126 \, x^{8} - 228 \, x^{4} + 2 \, {\left (468 \, x^{6} - 662 \, x^{2} + \sqrt {2} {\left (331 \, x^{6} - 468 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} \sqrt {-41 \, \sqrt {2} + 58} + \sqrt {2} {\left (91 \, x^{8} - 160 \, x^{4} + 46\right )} + 68\right )} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}}}{x^{8} - 2}\right ) + \frac {1}{32} \, \sqrt {2} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} + \sqrt {2} {\left (6 \, x^{5} - 11 \, x\right )} - 12 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (194 \, x^{7} - 274 \, x^{3} + \sqrt {2} {\left (137 \, x^{7} - 194 \, x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} \sqrt {-41 \, \sqrt {2} + 58} - {\left (126 \, x^{8} - 228 \, x^{4} + 2 \, {\left (468 \, x^{6} - 662 \, x^{2} + \sqrt {2} {\left (331 \, x^{6} - 468 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} \sqrt {-41 \, \sqrt {2} + 58} + \sqrt {2} {\left (91 \, x^{8} - 160 \, x^{4} + 46\right )} + 68\right )} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}}}{x^{8} - 2}\right ) + \frac {1}{32} \, \sqrt {2} {\left (41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} - \sqrt {2} {\left (6 \, x^{5} - 11 \, x\right )} - 12 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (194 \, x^{7} - 274 \, x^{3} - \sqrt {2} {\left (137 \, x^{7} - 194 \, x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} \sqrt {41 \, \sqrt {2} + 58} + {\left (126 \, x^{8} - 228 \, x^{4} + 2 \, {\left (468 \, x^{6} - 662 \, x^{2} - \sqrt {2} {\left (331 \, x^{6} - 468 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} \sqrt {41 \, \sqrt {2} + 58} - \sqrt {2} {\left (91 \, x^{8} - 160 \, x^{4} + 46\right )} + 68\right )} {\left (41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}}}{x^{8} - 2}\right ) - \frac {1}{32} \, \sqrt {2} {\left (41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} - \sqrt {2} {\left (6 \, x^{5} - 11 \, x\right )} - 12 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (194 \, x^{7} - 274 \, x^{3} - \sqrt {2} {\left (137 \, x^{7} - 194 \, x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} \sqrt {41 \, \sqrt {2} + 58} - {\left (126 \, x^{8} - 228 \, x^{4} + 2 \, {\left (468 \, x^{6} - 662 \, x^{2} - \sqrt {2} {\left (331 \, x^{6} - 468 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} \sqrt {41 \, \sqrt {2} + 58} - \sqrt {2} {\left (91 \, x^{8} - 160 \, x^{4} + 46\right )} + 68\right )} {\left (41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}}}{x^{8} - 2}\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 {x^{4} + 2}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{8} - 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 = -0.01 \begin {gather*} \int \frac {x^4+2}{{\left (x^4-1\right )}^{1/4}\,\left (x^8-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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