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 = 0.07, 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 = {1359, 1123, 1166, 203} \[ -\frac {1}{2 x^2}+\frac {1}{2} \sqrt {\frac {1}{5} \left (9-4 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x^2\right )-\frac {\left (3+\sqrt {5}\right )^{3/2} \tan ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^2\right )}{4 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 1123
Rule 1166
Rule 1359
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1+3 x^4+x^8\right )} \, dx &=\frac {1}{2} \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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] time = 0.02, size = 65, normalized size = 0.73 \[ -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8+3 \text {$\#$1}^4+1\& ,\frac {\text {$\#$1}^4 \log (x-\text {$\#$1})+3 \log (x-\text {$\#$1})}{2 \text {$\#$1}^6+3 \text {$\#$1}^2}\& \right ]-\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 158, normalized size = 1.78 \[ -\frac {2 \, \sqrt {5} x^{2} \sqrt {-4 \, \sqrt {5} + 9} \arctan \left (\frac {1}{4} \, \sqrt {2 \, x^{4} + \sqrt {5} + 3} {\left (\sqrt {5} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {5} + 9} - \frac {1}{2} \, {\left (\sqrt {5} x^{2} + 3 \, x^{2}\right )} \sqrt {-4 \, \sqrt {5} + 9}\right ) + 2 \, \sqrt {5} x^{2} \sqrt {4 \, \sqrt {5} + 9} \arctan \left (-\frac {1}{4} \, {\left (2 \, \sqrt {5} x^{2} - 6 \, x^{2} - \sqrt {2 \, x^{4} - \sqrt {5} + 3} {\left (\sqrt {5} \sqrt {2} - 3 \, \sqrt {2}\right )}\right )} \sqrt {4 \, \sqrt {5} + 9}\right ) + 5}{10 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 68, normalized size = 0.76 \[ -\frac {1}{20} \, {\left (x^{4} {\left (\sqrt {5} - 5\right )} + 3 \, \sqrt {5} - 15\right )} \arctan \left (\frac {2 \, x^{2}}{\sqrt {5} + 1}\right ) - \frac {1}{20} \, {\left (x^{4} {\left (\sqrt {5} + 5\right )} + 3 \, \sqrt {5} + 15\right )} \arctan \left (\frac {2 \, x^{2}}{\sqrt {5} - 1}\right ) - \frac {1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 117, normalized size = 1.31 \[ -\frac {\arctan \left (\frac {4 x^{2}}{-2+2 \sqrt {5}}\right )}{-2+2 \sqrt {5}}-\frac {3 \sqrt {5}\, \arctan \left (\frac {4 x^{2}}{-2+2 \sqrt {5}}\right )}{5 \left (-2+2 \sqrt {5}\right )}-\frac {\arctan \left (\frac {4 x^{2}}{2+2 \sqrt {5}}\right )}{2+2 \sqrt {5}}+\frac {3 \sqrt {5}\, \arctan \left (\frac {4 x^{2}}{2+2 \sqrt {5}}\right )}{5 \left (2+2 \sqrt {5}\right )}-\frac {1}{2 x^{2}} \]
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 \[ -\frac {1}{2 \, x^{2}} - \int \frac {{\left (x^{4} + 3\right )} x}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]
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 \[ 2\,\mathrm {atanh}\left (\frac {26880\,x^2\,\sqrt {-\frac {\sqrt {5}}{20}-\frac {9}{80}}}{3520\,\sqrt {5}+7872}+\frac {12032\,\sqrt {5}\,x^2\,\sqrt {-\frac {\sqrt {5}}{20}-\frac {9}{80}}}{3520\,\sqrt {5}+7872}\right )\,\sqrt {-\frac {\sqrt {5}}{20}-\frac {9}{80}}-2\,\mathrm {atanh}\left (\frac {26880\,x^2\,\sqrt {\frac {\sqrt {5}}{20}-\frac {9}{80}}}{3520\,\sqrt {5}-7872}-\frac {12032\,\sqrt {5}\,x^2\,\sqrt {\frac {\sqrt {5}}{20}-\frac {9}{80}}}{3520\,\sqrt {5}-7872}\right )\,\sqrt {\frac {\sqrt {5}}{20}-\frac {9}{80}}-\frac {1}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 56, normalized size = 0.63 \[ - 2 \left (\frac {\sqrt {5}}{10} + \frac {1}{4}\right ) \operatorname {atan}{\left (\frac {2 x^{2}}{-1 + \sqrt {5}} \right )} + 2 \left (\frac {1}{4} - \frac {\sqrt {5}}{10}\right ) \operatorname {atan}{\left (\frac {2 x^{2}}{1 + \sqrt {5}} \right )} - \frac {1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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