Optimal. Leaf size=140 \[ -\frac {1}{8} \log \left (x^2-x+1\right )+\frac {1}{8} \log \left (x^2+x+1\right )-\frac {\log \left (x^2-\sqrt {3} x+1\right )}{8 \sqrt {3}}+\frac {\log \left (x^2+\sqrt {3} x+1\right )}{8 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x\right )+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \tan ^{-1}\left (2 x+\sqrt {3}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1419, 1094, 634, 618, 204, 628} \begin {gather*} -\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}}-\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 {2 x+1}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \tan ^{-1}\left (2 x+\sqrt {3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1094
Rule 1419
Rubi steps
\begin {align*} \int \frac {1+x^4}{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\\ &=\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}-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}}\\ &=\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+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} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 x\right )-\frac {1}{4} \operatorname {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] time = 0.17, size = 135, normalized size = 0.96 \begin {gather*} \frac {1}{48} \left (-6 \log \left (x^2-x+1\right )+6 \log \left (x^2+x+1\right )+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 {2 x-1}{\sqrt {3}}\right )+4 \sqrt {3} \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^4}{1+x^4+x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.46, size = 211, normalized size = 1.51 \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}{3} \, \sqrt {6} \sqrt {3} \sqrt {\sqrt {6} \sqrt {2} x + 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 + \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 (\sqrt {6} \sqrt {2} x + 2 \, x^{2} + 2\right ) - \frac {1}{48} \, \sqrt {6} \sqrt {2} \log \left (-\sqrt {6} \sqrt {2} x + 2 \, x^{2} + 2\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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giac [A] time = 0.42, 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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maple [A] time = 0.02, size = 109, normalized size = 0.78 \begin {gather*} \frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\arctan \left (2 x -\sqrt {3}\right )}{4}+\frac {\arctan \left (2 x +\sqrt {3}\right )}{4}-\frac {\sqrt {3}\, \ln \left (x^{2}-\sqrt {3}\, x +1\right )}{24}+\frac {\sqrt {3}\, \ln \left (x^{2}+\sqrt {3}\, x +1\right )}{24}-\frac {\ln \left (x^{2}-x +1\right )}{8}+\frac {\ln \left (x^{2}+x +1\right )}{8} \end {gather*}
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*} \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}{2} \, \int \frac {1}{x^{4} - x^{2} + 1}\,{d x} + \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 = 0.14, size = 95, normalized size = 0.68 \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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sympy [C] time = 0.70, size = 190, normalized size = 1.36 \begin {gather*} \left (- \frac {1}{8} - \frac {\sqrt {3} i}{24}\right ) \log {\left (x - 1 - \frac {\sqrt {3} i}{3} + 9216 \left (- \frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{5} \right )} + \left (- \frac {1}{8} + \frac {\sqrt {3} i}{24}\right ) \log {\left (x - 1 + 9216 \left (- \frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{5} + \frac {\sqrt {3} i}{3} \right )} + \left (\frac {1}{8} - \frac {\sqrt {3} i}{24}\right ) \log {\left (x + 1 - \frac {\sqrt {3} i}{3} + 9216 \left (\frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{5} \right )} + \left (\frac {1}{8} + \frac {\sqrt {3} i}{24}\right ) \log {\left (x + 1 + 9216 \left (\frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{5} + \frac {\sqrt {3} i}{3} \right )} + \operatorname {RootSum} {\left (2304 t^{4} + 48 t^{2} + 1, \left (t \mapsto t \log {\left (9216 t^{5} + 8 t + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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