Optimal. Leaf size=67 \[ -\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (1-x+x^2\right )+\frac {1}{4} \log \left (1+x+x^2\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1108, 648, 632,
210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\text {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (x^2-x+1\right )+\frac {1}{4} \log \left (x^2+x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1108
Rubi steps
\begin {align*} \int \frac {1}{1+x^2+x^4} \, dx &=\frac {1}{2} \int \frac {1-x}{1-x+x^2} \, dx+\frac {1}{2} \int \frac {1+x}{1+x+x^2} \, dx\\ &=\frac {1}{4} \int \frac {1}{1-x+x^2} \, dx-\frac {1}{4} \int \frac {-1+2 x}{1-x+x^2} \, dx+\frac {1}{4} \int \frac {1}{1+x+x^2} \, dx+\frac {1}{4} \int \frac {1+2 x}{1+x+x^2} \, dx\\ &=-\frac {1}{4} \log \left (1-x+x^2\right )+\frac {1}{4} \log \left (1+x+x^2\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (1-x+x^2\right )+\frac {1}{4} \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.04, size = 73, normalized size = 1.09 \begin {gather*} \frac {i \left (\sqrt {1-i \sqrt {3}} \tan ^{-1}\left (\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )-\sqrt {1+i \sqrt {3}} \tan ^{-1}\left (\frac {1}{2} \left (i+\sqrt {3}\right ) x\right )\right )}{\sqrt {6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 54, normalized size = 0.81
method | result | size |
default | \(\frac {\ln \left (x^{2}+x +1\right )}{4}+\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}-\frac {\ln \left (x^{2}-x +1\right )}{4}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\) | \(54\) |
risch | \(-\frac {\ln \left (4 x^{2}-4 x +4\right )}{4}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}+\frac {\ln \left (4 x^{2}+4 x +4\right )}{4}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.88, size = 53, normalized size = 0.79 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.60, size = 53, normalized size = 0.79 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, \log \left (x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 70, normalized size = 1.04 \begin {gather*} - \frac {\log {\left (x^{2} - x + 1 \right )}}{4} + \frac {\log {\left (x^{2} + x + 1 \right )}}{4} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{6} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 53, normalized size = 0.79 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, \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 = 0.18, size = 47, normalized size = 0.70 \begin {gather*} \mathrm {atanh}\left (\frac {2\,x}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\mathrm {atanh}\left (\frac {2\,x}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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