Optimal. Leaf size=89 \[ -\frac {1}{2 x^2}+\frac {1}{2} \sqrt {\frac {1}{5} \left (9-4 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x^2\right )-\frac {\left (3+\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^2\right )}{4 \sqrt {10}} \]
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Rubi [A]
time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1373, 1137,
1180, 209} \begin {gather*} \frac {1}{2} \sqrt {\frac {1}{5} \left (9-4 \sqrt {5}\right )} \text {ArcTan}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x^2\right )-\frac {\left (3+\sqrt {5}\right )^{3/2} \text {ArcTan}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^2\right )}{4 \sqrt {10}}-\frac {1}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1137
Rule 1180
Rule 1373
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1+3 x^4+x^8\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (1+3 x^2+x^4\right )} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {1}{2} \text {Subst}\left (\int \frac {-3-x^2}{1+3 x^2+x^4} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {1}{20} \left (-5+3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx,x,x^2\right )-\frac {1}{20} \left (5+3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {1}{10} \sqrt {45-20 \sqrt {5}} \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x^2\right )-\frac {\left (3+\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^2\right )}{4 \sqrt {10}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.01, size = 65, normalized size = 0.73 \begin {gather*} -\frac {1}{2 x^2}-\frac {1}{4} \text {RootSum}\left [1+3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {3 \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{3 \text {$\#$1}^2+2 \text {$\#$1}^6}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 75, normalized size = 0.84
method | result | size |
risch | \(-\frac {1}{2 x^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}+90 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (35 \textit {\_R}^{3}+8 x^{2}+123 \textit {\_R} \right )\right )}{4}\) | \(42\) |
default | \(-\frac {1}{2 x^{2}}-\frac {\sqrt {5}\, \left (\sqrt {5}-3\right ) \arctan \left (\frac {4 x^{2}}{2 \sqrt {5}+2}\right )}{5 \left (2 \sqrt {5}+2\right )}-\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {4 x^{2}}{2 \sqrt {5}-2}\right )}{5 \left (2 \sqrt {5}-2\right )}\) | \(75\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs.
\(2 (53) = 106\).
time = 0.35, size = 144, normalized size = 1.62 \begin {gather*} -\frac {2 \, \sqrt {5} x^{2} \sqrt {-4 \, \sqrt {5} + 9} \arctan \left (\frac {1}{4} \, \sqrt {4 \, x^{4} + 2 \, \sqrt {5} + 6} {\left (\sqrt {5} + 3\right )} \sqrt {-4 \, \sqrt {5} + 9} - \frac {1}{2} \, {\left (\sqrt {5} x^{2} + 3 \, x^{2}\right )} \sqrt {-4 \, \sqrt {5} + 9}\right ) + 2 \, \sqrt {5} x^{2} \sqrt {4 \, \sqrt {5} + 9} \arctan \left (-\frac {1}{4} \, {\left (2 \, \sqrt {5} x^{2} - 6 \, x^{2} - \sqrt {4 \, x^{4} - 2 \, \sqrt {5} + 6} {\left (\sqrt {5} - 3\right )}\right )} \sqrt {4 \, \sqrt {5} + 9}\right ) + 5}{10 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 56, normalized size = 0.63 \begin {gather*} - 2 \left (\frac {\sqrt {5}}{10} + \frac {1}{4}\right ) \operatorname {atan}{\left (\frac {2 x^{2}}{-1 + \sqrt {5}} \right )} + 2 \cdot \left (\frac {1}{4} - \frac {\sqrt {5}}{10}\right ) \operatorname {atan}{\left (\frac {2 x^{2}}{1 + \sqrt {5}} \right )} - \frac {1}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.32, size = 68, normalized size = 0.76 \begin {gather*} -\frac {1}{20} \, {\left (x^{4} {\left (\sqrt {5} - 5\right )} + 3 \, \sqrt {5} - 15\right )} \arctan \left (\frac {2 \, x^{2}}{\sqrt {5} + 1}\right ) - \frac {1}{20} \, {\left (x^{4} {\left (\sqrt {5} + 5\right )} + 3 \, \sqrt {5} + 15\right )} \arctan \left (\frac {2 \, x^{2}}{\sqrt {5} - 1}\right ) - \frac {1}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.30, size = 130, normalized size = 1.46 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {26880\,x^2\,\sqrt {-\frac {\sqrt {5}}{20}-\frac {9}{80}}}{3520\,\sqrt {5}+7872}+\frac {12032\,\sqrt {5}\,x^2\,\sqrt {-\frac {\sqrt {5}}{20}-\frac {9}{80}}}{3520\,\sqrt {5}+7872}\right )\,\sqrt {-\frac {\sqrt {5}}{20}-\frac {9}{80}}-2\,\mathrm {atanh}\left (\frac {26880\,x^2\,\sqrt {\frac {\sqrt {5}}{20}-\frac {9}{80}}}{3520\,\sqrt {5}-7872}-\frac {12032\,\sqrt {5}\,x^2\,\sqrt {\frac {\sqrt {5}}{20}-\frac {9}{80}}}{3520\,\sqrt {5}-7872}\right )\,\sqrt {\frac {\sqrt {5}}{20}-\frac {9}{80}}-\frac {1}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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