Optimal. Leaf size=54 \[ -\frac {1}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
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Rubi [A]
time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1373, 1137,
1175, 632, 210} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 1137
Rule 1175
Rule 1373
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (1+x^4+x^8\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (1+x^2+x^4\right )} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {1}{2} \text {Subst}\left (\int \frac {-1-x^2}{1+x^2+x^4} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^2\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^2\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.03, size = 100, normalized size = 1.85 \begin {gather*} \frac {1}{12} \left (-\frac {6}{x^2}-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 )+i \sqrt {3} \log \left (-1-i \sqrt {3}+2 x^2\right )-i \sqrt {3} \log \left (-1+i \sqrt {3}+2 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 57, normalized size = 1.06
method | result | size |
risch | \(-\frac {1}{2 x^{2}}-\frac {\sqrt {3}\, \arctan \left (\frac {x^{2} \sqrt {3}}{3}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {x^{6} \sqrt {3}}{3}+\frac {2 x^{2} \sqrt {3}}{3}\right )}{6}\) | \(44\) |
default | \(\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}-\frac {1}{2 x^{2}}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 42, normalized size = 0.78 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) - \frac {1}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 45, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {3} x^{2} \arctan \left (\frac {1}{3} \, \sqrt {3} x^{2}\right ) + \sqrt {3} x^{2} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x^{6} + 2 \, x^{2}\right )}\right ) + 3}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 53, normalized size = 0.98 \begin {gather*} \frac {\sqrt {3} \left (- 2 \operatorname {atan}{\left (\frac {\sqrt {3} x^{2}}{3} \right )} - 2 \operatorname {atan}{\left (\frac {\sqrt {3} x^{6}}{3} + \frac {2 \sqrt {3} x^{2}}{3} \right )}\right )}{12} - \frac {1}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.05, size = 42, normalized size = 0.78 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) - \frac {1}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 43, normalized size = 0.80 \begin {gather*} -\frac {\sqrt {3}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {3}\,x^6}{3}+\frac {2\,\sqrt {3}\,x^2}{3}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {3}\,x^2}{3}\right )\right )}{12}-\frac {1}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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