Optimal. Leaf size=85 \[ -\frac {\log \left (x^2-\sqrt {3} x+1\right )}{4 \sqrt {3}}+\frac {\log \left (x^2+\sqrt {3} x+1\right )}{4 \sqrt {3}}-\frac {1}{x}+\frac {1}{6} \tan ^{-1}\left (\sqrt {3}-2 x\right )-\frac {1}{3} \tan ^{-1}(x)-\frac {1}{6} \tan ^{-1}\left (2 x+\sqrt {3}\right ) \]
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Rubi [A] time = 0.25, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {325, 295, 634, 618, 204, 628, 203} \[ -\frac {\log \left (x^2-\sqrt {3} x+1\right )}{4 \sqrt {3}}+\frac {\log \left (x^2+\sqrt {3} x+1\right )}{4 \sqrt {3}}-\frac {1}{x}+\frac {1}{6} \tan ^{-1}\left (\sqrt {3}-2 x\right )-\frac {1}{3} \tan ^{-1}(x)-\frac {1}{6} \tan ^{-1}\left (2 x+\sqrt {3}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 295
Rule 325
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (1+x^6\right )} \, dx &=-\frac {1}{x}-\int \frac {x^4}{1+x^6} \, dx\\ &=-\frac {1}{x}-\frac {1}{3} \int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx-\frac {1}{3} \int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx-\frac {1}{3} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {1}{x}-\frac {1}{3} \tan ^{-1}(x)-\frac {1}{12} \int \frac {1}{1-\sqrt {3} x+x^2} \, dx-\frac {1}{12} \int \frac {1}{1+\sqrt {3} x+x^2} \, dx-\frac {\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx}{4 \sqrt {3}}+\frac {\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx}{4 \sqrt {3}}\\ &=-\frac {1}{x}-\frac {1}{3} \tan ^{-1}(x)-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{4 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{4 \sqrt {3}}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 x\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 x\right )\\ &=-\frac {1}{x}+\frac {1}{6} \tan ^{-1}\left (\sqrt {3}-2 x\right )-\frac {1}{3} \tan ^{-1}(x)-\frac {1}{6} \tan ^{-1}\left (\sqrt {3}+2 x\right )-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{4 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{4 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 82, normalized size = 0.96 \[ -\frac {\sqrt {3} x \log \left (x^2-\sqrt {3} x+1\right )-\sqrt {3} x \log \left (x^2+\sqrt {3} x+1\right )-2 x \tan ^{-1}\left (\sqrt {3}-2 x\right )+4 x \tan ^{-1}(x)+2 x \tan ^{-1}\left (2 x+\sqrt {3}\right )+12}{12 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 99, normalized size = 1.16 \[ \frac {\sqrt {3} x \log \left (x^{2} + \sqrt {3} x + 1\right ) - \sqrt {3} x \log \left (x^{2} - \sqrt {3} x + 1\right ) - 4 \, x \arctan \relax (x) + 4 \, x \arctan \left (-2 \, x + \sqrt {3} + 2 \, \sqrt {x^{2} - \sqrt {3} x + 1}\right ) + 4 \, x \arctan \left (-2 \, x - \sqrt {3} + 2 \, \sqrt {x^{2} + \sqrt {3} x + 1}\right ) - 12}{12 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 65, normalized size = 0.76 \[ \frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) - \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) - \frac {1}{x} - \frac {1}{6} \, \arctan \left (2 \, x + \sqrt {3}\right ) - \frac {1}{6} \, \arctan \left (2 \, x - \sqrt {3}\right ) - \frac {1}{3} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 66, normalized size = 0.78 \[ -\frac {\arctan \relax (x )}{3}-\frac {\arctan \left (2 x -\sqrt {3}\right )}{6}-\frac {\arctan \left (2 x +\sqrt {3}\right )}{6}-\frac {\sqrt {3}\, \ln \left (x^{2}-\sqrt {3}\, x +1\right )}{12}+\frac {\sqrt {3}\, \ln \left (x^{2}+\sqrt {3}\, x +1\right )}{12}-\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.41, size = 65, normalized size = 0.76 \[ \frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) - \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) - \frac {1}{x} - \frac {1}{6} \, \arctan \left (2 \, x + \sqrt {3}\right ) - \frac {1}{6} \, \arctan \left (2 \, x - \sqrt {3}\right ) - \frac {1}{3} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 56, normalized size = 0.66 \[ -\frac {\mathrm {atan}\relax (x)}{3}+\mathrm {atan}\left (\frac {2\,x}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\mathrm {atan}\left (\frac {2\,x}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.50, size = 71, normalized size = 0.84 \[ - \frac {\sqrt {3} \log {\left (x^{2} - \sqrt {3} x + 1 \right )}}{12} + \frac {\sqrt {3} \log {\left (x^{2} + \sqrt {3} x + 1 \right )}}{12} - \frac {\operatorname {atan}{\relax (x )}}{3} - \frac {\operatorname {atan}{\left (2 x - \sqrt {3} \right )}}{6} - \frac {\operatorname {atan}{\left (2 x + \sqrt {3} \right )}}{6} - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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