Optimal. Leaf size=75 \[ -\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{8} \log \left (x^4-x^2+1\right )-\frac {1}{8} \log \left (x^4+x^2+1\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1359, 1127, 1161, 618, 204, 1164, 628} \[ \frac {1}{8} \log \left (x^4-x^2+1\right )-\frac {1}{8} \log \left (x^4+x^2+1\right )-\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 1127
Rule 1161
Rule 1164
Rule 1359
Rubi steps
\begin {align*} \int \frac {x^5}{1+x^4+x^8} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{1+x^2+x^4} \, dx,x,x^2\right )\\ &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^2+x^4} \, dx,x,x^2\right )\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^2+x^4} \, dx,x,x^2\right )\\ &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {1+2 x}{-1-x-x^2} \, dx,x,x^2\right )+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1-2 x}{-1+x-x^2} \, dx,x,x^2\right )+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^2\right )+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{8} \log \left (1-x^2+x^4\right )-\frac {1}{8} \log \left (1+x^2+x^4\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^2\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^2\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{8} \log \left (1-x^2+x^4\right )-\frac {1}{8} \log \left (1+x^2+x^4\right )\\ \end {align*}
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Mathematica [C] time = 0.12, size = 94, normalized size = 1.25 \[ \frac {\sqrt {1-i \sqrt {3}} \left (\sqrt {3}-i\right ) \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}-i\right ) x^2\right )+\sqrt {1+i \sqrt {3}} \left (\sqrt {3}+i\right ) \tan ^{-1}\left (\frac {1}{2} \left (\sqrt {3}+i\right ) x^2\right )}{4 \sqrt {6}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.87, size = 61, normalized size = 0.81 \[ \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{4} + x^{2} + 1\right ) + \frac {1}{8} \, \log \left (x^{4} - x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 61, normalized size = 0.81 \[ \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{4} + x^{2} + 1\right ) + \frac {1}{8} \, \log \left (x^{4} - x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 62, normalized size = 0.83 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}+1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\ln \left (x^{4}-x^{2}+1\right )}{8}-\frac {\ln \left (x^{4}+x^{2}+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.49, size = 61, normalized size = 0.81 \[ \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{4} + x^{2} + 1\right ) + \frac {1}{8} \, \log \left (x^{4} - x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 51, normalized size = 0.68 \[ \mathrm {atanh}\left (\frac {2\,x^2}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\mathrm {atanh}\left (\frac {2\,x^2}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 76, normalized size = 1.01 \[ \frac {\log {\left (x^{4} - x^{2} + 1 \right )}}{8} - \frac {\log {\left (x^{4} + x^{2} + 1 \right )}}{8} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{2}}{3} - \frac {\sqrt {3}}{3} \right )}}{12} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{2}}{3} + \frac {\sqrt {3}}{3} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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