Optimal. Leaf size=140 \[ -\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}+2 x\right )+\frac {1}{8} \log \left (1-x+x^2\right )-\frac {1}{8} \log \left (1+x+x^2\right )-\frac {1}{8} \sqrt {3} \log \left (1-\sqrt {3} x+x^2\right )+\frac {1}{8} \sqrt {3} \log \left (1+\sqrt {3} x+x^2\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1435, 1183,
648, 632, 210, 642} \begin {gather*} -\frac {1}{4} \sqrt {3} \text {ArcTan}\left (\frac {1-2 x}{\sqrt {3}}\right )+\frac {1}{4} \text {ArcTan}\left (\sqrt {3}-2 x\right )+\frac {1}{4} \sqrt {3} \text {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )-\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 {1}{8} \sqrt {3} \log \left (x^2-\sqrt {3} x+1\right )+\frac {1}{8} \sqrt {3} \log \left (x^2+\sqrt {3} x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1183
Rule 1435
Rubi steps
\begin {align*} \int \frac {1-x^4}{1+x^4+x^8} \, dx &=-\left (\frac {1}{2} \int \frac {1+2 x^2}{-1-x^2-x^4} \, dx\right )-\frac {1}{2} \int \frac {1-2 x^2}{-1+x^2-x^4} \, dx\\ &=\frac {1}{4} \int \frac {1+x}{1-x+x^2} \, dx+\frac {1}{4} \int \frac {1-x}{1+x+x^2} \, dx+\frac {\int \frac {\sqrt {3}-3 x}{1-\sqrt {3} x+x^2} \, dx}{4 \sqrt {3}}+\frac {\int \frac {\sqrt {3}+3 x}{1+\sqrt {3} x+x^2} \, dx}{4 \sqrt {3}}\\ &=\frac {1}{8} \int \frac {-1+2 x}{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 {3}{8} \int \frac {1}{1-x+x^2} \, dx+\frac {3}{8} \int \frac {1}{1+x+x^2} \, dx-\frac {1}{8} \sqrt {3} \int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx+\frac {1}{8} \sqrt {3} \int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx\\ &=\frac {1}{8} \log \left (1-x+x^2\right )-\frac {1}{8} \log \left (1+x+x^2\right )-\frac {1}{8} \sqrt {3} \log \left (1-\sqrt {3} x+x^2\right )+\frac {1}{8} \sqrt {3} \log \left (1+\sqrt {3} x+x^2\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 {3}{4} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )-\frac {3}{4} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=-\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-\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 {1}{8} \sqrt {3} \log \left (1-\sqrt {3} x+x^2\right )+\frac {1}{8} \sqrt {3} \log \left (1+\sqrt {3} x+x^2\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.10, size = 129, normalized size = 0.92 \begin {gather*} \frac {1}{8} \left (-2 \sqrt {-2-2 i \sqrt {3}} \tan ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) x\right )-2 \sqrt {-2+2 i \sqrt {3}} \tan ^{-1}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ) x\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )+\log \left (1-x+x^2\right )-\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 | \(\frac {\ln \left (4 x^{2}-4 x +4\right )}{8}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{4}-\frac {\ln \left (4 x^{2}+4 x +4\right )}{8}+\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{4}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-\textit {\_R}^{3}+\textit {\_R} +x \right )\right )}{4}\) | \(87\) |
default | \(-\frac {\ln \left (x^{2}+x +1\right )}{8}+\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{4}-\frac {\ln \left (1+x^{2}-x \sqrt {3}\right ) \sqrt {3}}{8}-\frac {\arctan \left (2 x -\sqrt {3}\right )}{4}+\frac {\ln \left (1+x^{2}+x \sqrt {3}\right ) \sqrt {3}}{8}-\frac {\arctan \left (2 x +\sqrt {3}\right )}{4}+\frac {\ln \left (x^{2}-x +1\right )}{8}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{4}\) | \(109\) |
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 = 0.36, size = 142, normalized size = 1.01 \begin {gather*} \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{8} \, \sqrt {3} \log \left (16 \, x^{2} + 16 \, \sqrt {3} x + 16\right ) - \frac {1}{8} \, \sqrt {3} \log \left (16 \, x^{2} - 16 \, \sqrt {3} x + 16\right ) + \frac {1}{2} \, \arctan \left (-2 \, x + \sqrt {3} + 2 \, \sqrt {x^{2} - \sqrt {3} x + 1}\right ) + \frac {1}{2} \, \arctan \left (-2 \, x - \sqrt {3} + 2 \, \sqrt {x^{2} + \sqrt {3} x + 1}\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 = 0.32, size = 148, normalized size = 1.06 \begin {gather*} - \left (- \frac {1}{8} - \frac {\sqrt {3} i}{8}\right ) \log {\left (x + 1024 \left (- \frac {1}{8} - \frac {\sqrt {3} i}{8}\right )^{5} \right )} - \left (- \frac {1}{8} + \frac {\sqrt {3} i}{8}\right ) \log {\left (x + 1024 \left (- \frac {1}{8} + \frac {\sqrt {3} i}{8}\right )^{5} \right )} - \left (\frac {1}{8} - \frac {\sqrt {3} i}{8}\right ) \log {\left (x + 1024 \left (\frac {1}{8} - \frac {\sqrt {3} i}{8}\right )^{5} \right )} - \left (\frac {1}{8} + \frac {\sqrt {3} i}{8}\right ) \log {\left (x + 1024 \left (\frac {1}{8} + \frac {\sqrt {3} i}{8}\right )^{5} \right )} - \operatorname {RootSum} {\left (256 t^{4} - 16 t^{2} + 1, \left ( t \mapsto t \log {\left (1024 t^{5} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.21, size = 108, normalized size = 0.77 \begin {gather*} \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{8} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) - \frac {1}{8} \, \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 = 0.19, size = 109, normalized size = 0.78 \begin {gather*} -\mathrm {atan}\left (\frac {54\,\sqrt {3}\,x}{-81+\sqrt {3}\,27{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{4}+\frac {1}{4}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {54\,\sqrt {3}\,x}{81+\sqrt {3}\,27{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{4}-\frac {1}{4}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\sqrt {3}\,x\,54{}\mathrm {i}}{-81+\sqrt {3}\,27{}\mathrm {i}}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )-\mathrm {atan}\left (\frac {\sqrt {3}\,x\,54{}\mathrm {i}}{81+\sqrt {3}\,27{}\mathrm {i}}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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