Optimal. Leaf size=140 \[ \frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x\right )-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}+2 x\right )+\frac {1}{8} \log \left (1-x+x^2\right )-\frac {1}{8} \log \left (1+x+x^2\right )-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{8 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{8 \sqrt {3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1387, 1108,
648, 632, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \text {ArcTan}\left (\sqrt {3}-2 x\right )-\frac {\text {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \text {ArcTan}\left (2 x+\sqrt {3}\right )+\frac {1}{8} \log \left (x^2-x+1\right )-\frac {1}{8} \log \left (x^2+x+1\right )-\frac {\log \left (x^2-\sqrt {3} x+1\right )}{8 \sqrt {3}}+\frac {\log \left (x^2+\sqrt {3} x+1\right )}{8 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1108
Rule 1387
Rubi steps
\begin {align*} \int \frac {x^2}{1+x^4+x^8} \, dx &=\frac {1}{2} \int \frac {1}{1-x^2+x^4} \, dx-\frac {1}{2} \int \frac {1}{1+x^2+x^4} \, dx\\ &=-\left (\frac {1}{4} \int \frac {1-x}{1-x+x^2} \, dx\right )-\frac {1}{4} \int \frac {1+x}{1+x+x^2} \, dx+\frac {\int \frac {\sqrt {3}-x}{1-\sqrt {3} x+x^2} \, dx}{4 \sqrt {3}}+\frac {\int \frac {\sqrt {3}+x}{1+\sqrt {3} x+x^2} \, dx}{4 \sqrt {3}}\\ &=-\left (\frac {1}{8} \int \frac {1}{1-x+x^2} \, dx\right )+\frac {1}{8} \int \frac {-1+2 x}{1-x+x^2} \, dx-\frac {1}{8} \int \frac {1}{1+x+x^2} \, dx-\frac {1}{8} \int \frac {1+2 x}{1+x+x^2} \, dx+\frac {1}{8} \int \frac {1}{1-\sqrt {3} x+x^2} \, dx+\frac {1}{8} \int \frac {1}{1+\sqrt {3} x+x^2} \, dx-\frac {\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx}{8 \sqrt {3}}+\frac {\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx}{8 \sqrt {3}}\\ &=\frac {1}{8} \log \left (1-x+x^2\right )-\frac {1}{8} \log \left (1+x+x^2\right )-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{8 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{8 \sqrt {3}}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 x\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 x\right )\\ &=\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x\right )-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}+2 x\right )+\frac {1}{8} \log \left (1-x+x^2\right )-\frac {1}{8} \log \left (1+x+x^2\right )-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{8 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{8 \sqrt {3}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.09, size = 135, normalized size = 0.96 \begin {gather*} \frac {1}{48} \left (4 i \sqrt {-6-6 i \sqrt {3}} \tan ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) x\right )-4 i \sqrt {-6+6 i \sqrt {3}} \tan ^{-1}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ) x\right )-4 \sqrt {3} \tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )-4 \sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )+6 \log \left (1-x+x^2\right )-6 \log \left (1+x+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 109, normalized size = 0.78
method | result | size |
risch | \(\frac {\ln \left (4 x^{2}-4 x +4\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}-\frac {\ln \left (4 x^{2}+4 x +4\right )}{8}-\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (9 \textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-3 \textit {\_R}^{3}+\textit {\_R} +x \right )\right )}{4}\) | \(89\) |
default | \(-\frac {\ln \left (x^{2}+x +1\right )}{8}-\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}+\frac {\arctan \left (2 x -\sqrt {3}\right )}{4}+\frac {\arctan \left (2 x +\sqrt {3}\right )}{4}-\frac {\ln \left (1+x^{2}-x \sqrt {3}\right ) \sqrt {3}}{24}+\frac {\ln \left (1+x^{2}+x \sqrt {3}\right ) \sqrt {3}}{24}+\frac {\ln \left (x^{2}-x +1\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}\) | \(109\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 215, normalized size = 1.54 \begin {gather*} -\frac {1}{12} \, \sqrt {6} \sqrt {3} \sqrt {2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x + \frac {1}{36} \, \sqrt {6} \sqrt {3} \sqrt {2} \sqrt {-72 \, \sqrt {6} \sqrt {2} x + 144 \, x^{2} + 144} + \sqrt {3}\right ) - \frac {1}{12} \, \sqrt {6} \sqrt {3} \sqrt {2} \arctan \left (-\frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {2} x + \frac {1}{3} \, \sqrt {6} \sqrt {3} \sqrt {\sqrt {6} \sqrt {2} x + 2 \, x^{2} + 2} - \sqrt {3}\right ) + \frac {1}{48} \, \sqrt {6} \sqrt {2} \log \left (72 \, \sqrt {6} \sqrt {2} x + 144 \, x^{2} + 144\right ) - \frac {1}{48} \, \sqrt {6} \sqrt {2} \log \left (-72 \, \sqrt {6} \sqrt {2} x + 144 \, x^{2} + 144\right ) - \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{8} \, \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.37, size = 214, normalized size = 1.53 \begin {gather*} \left (- \frac {1}{8} - \frac {\sqrt {3} i}{24}\right ) \log {\left (x + 442368 \left (- \frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{7} - 192 \left (- \frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{3} \right )} + \left (- \frac {1}{8} + \frac {\sqrt {3} i}{24}\right ) \log {\left (x - 192 \left (- \frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{3} + 442368 \left (- \frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{7} \right )} + \left (\frac {1}{8} - \frac {\sqrt {3} i}{24}\right ) \log {\left (x + 442368 \left (\frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{7} - 192 \left (\frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{3} \right )} + \left (\frac {1}{8} + \frac {\sqrt {3} i}{24}\right ) \log {\left (x - 192 \left (\frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{3} + 442368 \left (\frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{7} \right )} + \operatorname {RootSum} {\left (2304 t^{4} + 48 t^{2} + 1, \left ( t \mapsto t \log {\left (442368 t^{7} - 192 t^{3} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.75, size = 108, normalized size = 0.77 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{24} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) - \frac {1}{24} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + \frac {1}{4} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{4} \, \arctan \left (2 \, x - \sqrt {3}\right ) - \frac {1}{8} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{8} \, \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.31, size = 97, normalized size = 0.69 \begin {gather*} \mathrm {atan}\left (\frac {2\,x}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\mathrm {atan}\left (\frac {2\,x}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\mathrm {atan}\left (\frac {x\,2{}\mathrm {i}}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{12}+\frac {1}{4}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {x\,2{}\mathrm {i}}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{12}-\frac {1}{4}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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