Optimal. Leaf size=57 \[ \frac {x^2}{2}+\frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{4 \sqrt {3}}-\frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{4 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 57, 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, 1122, 1164, 628} \[ \frac {x^2}{2}+\frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{4 \sqrt {3}}-\frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 628
Rule 1122
Rule 1164
Rule 1359
Rubi steps
\begin {align*} \int \frac {x^9}{1-x^4+x^8} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^4}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=\frac {x^2}{2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1-x^2}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=\frac {x^2}{2}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {3}+2 x}{-1-\sqrt {3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt {3}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {3}-2 x}{-1+\sqrt {3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt {3}}\\ &=\frac {x^2}{2}+\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 55, normalized size = 0.96 \[ \frac {1}{12} \left (6 x^2+\sqrt {3} \log \left (-x^4+\sqrt {3} x^2-1\right )-\sqrt {3} \log \left (x^4+\sqrt {3} x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 47, normalized size = 0.82 \[ \frac {1}{2} \, x^{2} + \frac {1}{12} \, \sqrt {3} \log \left (\frac {x^{8} + 5 \, x^{4} - 2 \, \sqrt {3} {\left (x^{6} + x^{2}\right )} + 1}{x^{8} - x^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 99, normalized size = 1.74 \[ \frac {1}{2} \, x^{2} + \frac {1}{4} \, {\left (x^{4} - 1\right )} \arctan \left (2 \, x^{2} + \sqrt {3}\right ) + \frac {1}{4} \, {\left (x^{4} - 1\right )} \arctan \left (2 \, x^{2} - \sqrt {3}\right ) + \frac {1}{24} \, {\left (\sqrt {3} x^{4} - \sqrt {3}\right )} \log \left (x^{4} + \sqrt {3} x^{2} + 1\right ) - \frac {1}{24} \, {\left (\sqrt {3} x^{4} - \sqrt {3}\right )} \log \left (x^{4} - \sqrt {3} x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 0.77 \[ \frac {x^{2}}{2}+\frac {\sqrt {3}\, \ln \left (x^{4}-\sqrt {3}\, x^{2}+1\right )}{12}-\frac {\sqrt {3}\, \ln \left (x^{4}+\sqrt {3}\, x^{2}+1\right )}{12} \]
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 \[ \frac {1}{2} \, x^{2} + \int \frac {{\left (x^{4} - 1\right )} x}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 29, normalized size = 0.51 \[ \frac {x^2}{2}-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,x^2}{9\,\left (\frac {2\,x^4}{9}+\frac {2}{9}\right )}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 48, normalized size = 0.84 \[ \frac {x^{2}}{2} + \frac {\sqrt {3} \log {\left (x^{4} - \sqrt {3} x^{2} + 1 \right )}}{12} - \frac {\sqrt {3} \log {\left (x^{4} + \sqrt {3} x^{2} + 1 \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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