Optimal. Leaf size=54 \[ \frac {x^2}{2}+\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
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Rubi [A] time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1359, 1122, 1161, 618, 204} \[ \frac {x^2}{2}+\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 1122
Rule 1161
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 {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^2\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^2\right )\\ &=\frac {x^2}{2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^2\right )\\ &=\frac {x^2}{2}+\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 98, normalized size = 1.81 \[ \frac {x^2}{2}-\frac {\left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}-i\right ) x^2\right )}{2 \sqrt {6+6 i \sqrt {3}}}-\frac {\left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}+i\right ) x^2\right )}{2 \sqrt {6-6 i \sqrt {3}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.05, size = 40, normalized size = 0.74 \[ \frac {1}{2} \, x^{2} - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x^{2}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x^{6} + 2 \, x^{2}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 42, normalized size = 0.78 \[ \frac {1}{2} \, x^{2} - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.80 \[ \frac {x^{2}}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}+1\right ) \sqrt {3}}{3}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.39, size = 42, normalized size = 0.78 \[ \frac {1}{2} \, x^{2} - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 43, normalized size = 0.80 \[ \frac {x^2}{2}-\frac {\sqrt {3}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {3}\,x^6}{3}+\frac {2\,\sqrt {3}\,x^2}{3}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {3}\,x^2}{3}\right )\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 51, normalized size = 0.94 \[ \frac {x^{2}}{2} + \frac {\sqrt {3} \left (- 2 \operatorname {atan}{\left (\frac {\sqrt {3} x^{2}}{3} \right )} - 2 \operatorname {atan}{\left (\frac {\sqrt {3} x^{6}}{3} + \frac {2 \sqrt {3} x^{2}}{3} \right )}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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