Optimal. Leaf size=41 \[ -\frac {\tan ^{-1}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\log (x)-\frac {1}{8} \log \left (1-x^4+x^8\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1371, 719, 29,
648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (x^8-x^4+1\right )+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 210
Rule 632
Rule 642
Rule 648
Rule 719
Rule 1371
Rubi steps
\begin {align*} \int \frac {1}{x \left (1-x^4+x^8\right )} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x \left (1-x+x^2\right )} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1-x}{1-x+x^2} \, dx,x,x^4\right )\\ &=\log (x)+\frac {1}{8} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^4\right )-\frac {1}{8} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,x^4\right )\\ &=\log (x)-\frac {1}{8} \log \left (1-x^4+x^8\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^4\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\log (x)-\frac {1}{8} \log \left (1-x^4+x^8\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.01, size = 55, normalized size = 1.34 \begin {gather*} \log (x)-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-1+2 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 35, normalized size = 0.85
method | result | size |
risch | \(\ln \left (x \right )-\frac {\ln \left (x^{8}-x^{4}+1\right )}{8}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{4}-\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{12}\) | \(33\) |
default | \(\ln \left (x \right )-\frac {\ln \left (x^{8}-x^{4}+1\right )}{8}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{4}-1\right ) \sqrt {3}}{3}\right )}{12}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 38, normalized size = 0.93 \begin {gather*} \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 34, normalized size = 0.83 \begin {gather*} \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 41, normalized size = 1.00 \begin {gather*} \log {\left (x \right )} - \frac {\log {\left (x^{8} - x^{4} + 1 \right )}}{8} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} - \frac {\sqrt {3}}{3} \right )}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.19, size = 38, normalized size = 0.93 \begin {gather*} \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.29, size = 36, normalized size = 0.88 \begin {gather*} \ln \left (x\right )-\frac {\ln \left (x^8-x^4+1\right )}{8}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}-\frac {2\,\sqrt {3}\,x^4}{3}\right )}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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