Optimal. Leaf size=82 \[ -\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x^2\right )+\frac {1}{4} \tan ^{-1}\left (2 x^2+\sqrt {3}\right )-\frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{8 \sqrt {3}}+\frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{8 \sqrt {3}} \]
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Rubi [A] time = 0.06, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1359, 1094, 634, 618, 204, 628} \[ -\frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{8 \sqrt {3}}+\frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{8 \sqrt {3}}-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x^2\right )+\frac {1}{4} \tan ^{-1}\left (2 x^2+\sqrt {3}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1094
Rule 1359
Rubi steps
\begin {align*} \int \frac {x}{1-x^4+x^8} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {3}-x}{1-\sqrt {3} x+x^2} \, dx,x,x^2\right )}{4 \sqrt {3}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {3}+x}{1+\sqrt {3} x+x^2} \, dx,x,x^2\right )}{4 \sqrt {3}}\\ &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,x^2\right )+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,x^2\right )-\frac {\operatorname {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,x^2\right )}{8 \sqrt {3}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,x^2\right )}{8 \sqrt {3}}\\ &=-\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{8 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{8 \sqrt {3}}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 x^2\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 x^2\right )\\ &=-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x^2\right )+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}+2 x^2\right )-\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{8 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{8 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 83, normalized size = 1.01 \[ \frac {i \left (\sqrt {-1-i \sqrt {3}} \tan ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) x^2\right )-\sqrt {-1+i \sqrt {3}} \tan ^{-1}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ) x^2\right )\right )}{2 \sqrt {6}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.00, size = 171, normalized size = 2.09 \[ -\frac {1}{12} \, \sqrt {6} \sqrt {3} \sqrt {2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x^{2} + \frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2 \, x^{4} + \sqrt {6} \sqrt {2} x^{2} + 2} - \sqrt {3}\right ) - \frac {1}{12} \, \sqrt {6} \sqrt {3} \sqrt {2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x^{2} + \frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2 \, x^{4} - \sqrt {6} \sqrt {2} x^{2} + 2} + \sqrt {3}\right ) + \frac {1}{48} \, \sqrt {6} \sqrt {2} \log \left (2 \, x^{4} + \sqrt {6} \sqrt {2} x^{2} + 2\right ) - \frac {1}{48} \, \sqrt {6} \sqrt {2} \log \left (2 \, x^{4} - \sqrt {6} \sqrt {2} x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 64, normalized size = 0.78 \[ \frac {1}{24} \, \sqrt {3} \log \left (x^{4} + \sqrt {3} x^{2} + 1\right ) - \frac {1}{24} \, \sqrt {3} \log \left (x^{4} - \sqrt {3} x^{2} + 1\right ) + \frac {1}{4} \, \arctan \left (2 \, x^{2} + \sqrt {3}\right ) + \frac {1}{4} \, \arctan \left (2 \, x^{2} - \sqrt {3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 0.79 \[ \frac {\arctan \left (2 x^{2}-\sqrt {3}\right )}{4}+\frac {\arctan \left (2 x^{2}+\sqrt {3}\right )}{4}-\frac {\sqrt {3}\, \ln \left (x^{4}-\sqrt {3}\, x^{2}+1\right )}{24}+\frac {\sqrt {3}\, \ln \left (x^{4}+\sqrt {3}\, x^{2}+1\right )}{24} \]
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 \[ \int \frac {x}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 53, normalized size = 0.65 \[ -\mathrm {atan}\left (-\frac {x^2}{2}+\frac {\sqrt {3}\,x^2\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\mathrm {atan}\left (\frac {x^2}{2}+\frac {\sqrt {3}\,x^2\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 70, normalized size = 0.85 \[ - \frac {\sqrt {3} \log {\left (x^{4} - \sqrt {3} x^{2} + 1 \right )}}{24} + \frac {\sqrt {3} \log {\left (x^{4} + \sqrt {3} x^{2} + 1 \right )}}{24} + \frac {\operatorname {atan}{\left (2 x^{2} - \sqrt {3} \right )}}{4} + \frac {\operatorname {atan}{\left (2 x^{2} + \sqrt {3} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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