Optimal. Leaf size=147 \[ -\frac {1}{3 x^3}+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x\right )-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}+2 x\right )+\frac {1}{8} \log \left (1-x+x^2\right )-\frac {1}{8} \log \left (1+x+x^2\right )+\frac {\log \left (1-\sqrt {3} x+x^2\right )}{8 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x+x^2\right )}{8 \sqrt {3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1382, 1433,
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}{3 x^3}+\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 1382
Rule 1433
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (1+x^4+x^8\right )} \, dx &=-\frac {1}{3 x^3}+\frac {1}{3} \int \frac {-3-3 x^4}{1+x^4+x^8} \, dx\\ &=-\frac {1}{3 x^3}-\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}{3 x^3}-\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}{3 x^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}{3 x^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 {1}{3 x^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 {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 = 0.17, size = 148, normalized size = 1.01 \begin {gather*} \frac {1}{24} \left (-\frac {8}{x^3}-\frac {4 i \tan ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) x\right )}{\sqrt {\frac {1}{6} i \left (i+\sqrt {3}\right )}}+\frac {4 i \tan ^{-1}\left (\frac {1}{2} \left (1+i \sqrt {3}\right ) x\right )}{\sqrt {-\frac {1}{6} i \left (-i+\sqrt {3}\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 )+3 \log \left (1-x+x^2\right )-3 \log \left (1+x+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 114, normalized size = 0.78
method | result | size |
risch | \(-\frac {1}{3 x^{3}}-\frac {\ln \left (4 x^{2}+4 x +4\right )}{8}-\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}+\frac {\ln \left (4 x^{2}-4 x +4\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (9 \textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (3 \textit {\_R}^{3}-\textit {\_R} +x \right )\right )}{4}\) | \(96\) |
default | \(-\frac {\ln \left (x^{2}+x +1\right )}{8}-\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}-\frac {1}{3 x^{3}}-\frac {\arctan \left (2 x -\sqrt {3}\right )}{4}-\frac {\arctan \left (2 x +\sqrt {3}\right )}{4}+\frac {\ln \left (1+x^{2}-x \sqrt {3}\right ) \sqrt {3}}{24}-\frac {\ln \left (1+x^{2}+x \sqrt {3}\right ) \sqrt {3}}{24}+\frac {\ln \left (x^{2}-x +1\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}\) | \(114\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 244 vs.
\(2 (113) = 226\).
time = 0.38, size = 244, normalized size = 1.66 \begin {gather*} \frac {4 \, \sqrt {6} \sqrt {3} \sqrt {2} x^{3} \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 ) + 4 \, \sqrt {6} \sqrt {3} \sqrt {2} x^{3} \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 ) - \sqrt {6} \sqrt {2} x^{3} \log \left (72 \, \sqrt {6} \sqrt {2} x + 144 \, x^{2} + 144\right ) + \sqrt {6} \sqrt {2} x^{3} \log \left (-72 \, \sqrt {6} \sqrt {2} x + 144 \, x^{2} + 144\right ) - 4 \, \sqrt {3} x^{3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 4 \, \sqrt {3} x^{3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 6 \, x^{3} \log \left (x^{2} + x + 1\right ) + 6 \, x^{3} \log \left (x^{2} - x + 1\right ) - 16}{48 \, x^{3}} \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 = 0.42, size = 197, normalized size = 1.34 \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 )} - \frac {1}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.13, size = 113, 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}{3 \, x^{3}} - \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 = 0.03, size = 104, normalized size = 0.71 \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 )-\frac {1}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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