Optimal. Leaf size=57 \[ -\frac {1}{2 x^2}-\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1373, 1137,
1178, 642} \begin {gather*} -\frac {1}{2 x^2}-\frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{4 \sqrt {3}}+\frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 642
Rule 1137
Rule 1178
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 {\text {Subst}\left (\int \frac {\sqrt {3}+2 x}{-1-\sqrt {3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {\sqrt {3}-2 x}{-1+\sqrt {3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt {3}}\\ &=-\frac {1}{2 x^2}-\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 55, normalized size = 0.96 \begin {gather*} \frac {1}{12} \left (-\frac {6}{x^2}-\sqrt {3} \log \left (-1+\sqrt {3} x^2-x^4\right )+\sqrt {3} \log \left (1+\sqrt {3} x^2+x^4\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 44, normalized size = 0.77
method | result | size |
default | \(-\frac {1}{2 x^{2}}-\frac {\ln \left (1+x^{4}-x^{2} \sqrt {3}\right ) \sqrt {3}}{12}+\frac {\ln \left (1+x^{4}+x^{2} \sqrt {3}\right ) \sqrt {3}}{12}\) | \(44\) |
risch | \(-\frac {1}{2 x^{2}}-\frac {\ln \left (1+x^{4}-x^{2} \sqrt {3}\right ) \sqrt {3}}{12}+\frac {\ln \left (1+x^{4}+x^{2} \sqrt {3}\right ) \sqrt {3}}{12}\) | \(44\) |
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 = 0.35, size = 50, normalized size = 0.88 \begin {gather*} \frac {\sqrt {3} x^{2} \log \left (\frac {x^{8} + 5 \, x^{4} + 2 \, \sqrt {3} {\left (x^{6} + x^{2}\right )} + 1}{x^{8} - x^{4} + 1}\right ) - 6}{12 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 49, normalized size = 0.86 \begin {gather*} - \frac {\sqrt {3} \log {\left (x^{4} - \sqrt {3} x^{2} + 1 \right )}}{12} + \frac {\sqrt {3} \log {\left (x^{4} + \sqrt {3} x^{2} + 1 \right )}}{12} - \frac {1}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (43) = 86\).
time = 3.35, size = 99, normalized size = 1.74 \begin {gather*} -\frac {1}{4} \, {\left (x^{4} - 1\right )} \arctan \left (2 \, x^{2} + \sqrt {3}\right ) - \frac {1}{4} \, {\left (x^{4} - 1\right )} \arctan \left (2 \, x^{2} - \sqrt {3}\right ) - \frac {1}{24} \, {\left (\sqrt {3} x^{4} - \sqrt {3}\right )} \log \left (x^{4} + \sqrt {3} x^{2} + 1\right ) + \frac {1}{24} \, {\left (\sqrt {3} x^{4} - \sqrt {3}\right )} \log \left (x^{4} - \sqrt {3} x^{2} + 1\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 = 1.27, size = 29, normalized size = 0.51 \begin {gather*} \frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,x^2}{9\,\left (\frac {2\,x^4}{9}+\frac {2}{9}\right )}\right )}{6}-\frac {1}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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