Optimal. Leaf size=37 \[ -\frac {\tan ^{-1}\left (\frac {1+2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{8} \log \left (1+x^4+x^8\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1371, 648, 632,
210, 642} \begin {gather*} \frac {1}{8} \log \left (x^8+x^4+1\right )-\frac {\text {ArcTan}\left (\frac {2 x^4+1}{\sqrt {3}}\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1371
Rubi steps
\begin {align*} \int \frac {x^7}{1+x^4+x^8} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{1+x+x^2} \, dx,x,x^4\right )\\ &=-\left (\frac {1}{8} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^4\right )\right )+\frac {1}{8} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,x^4\right )\\ &=\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}}+\frac {1}{8} \log \left (1+x^4+x^8\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 37, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {1+2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{8} \log \left (1+x^4+x^8\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 31, normalized size = 0.84
method | result | size |
default | \(\frac {\ln \left (x^{8}+x^{4}+1\right )}{8}-\frac {\arctan \left (\frac {\left (2 x^{4}+1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}\) | \(31\) |
risch | \(\frac {\ln \left (4 x^{8}+4 x^{4}+4\right )}{8}-\frac {\arctan \left (\frac {\left (2 x^{4}+1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 30, normalized size = 0.81 \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 30, normalized size = 0.81 \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 37, normalized size = 1.00 \begin {gather*} \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 = 5.36, size = 30, normalized size = 0.81 \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 32, normalized size = 0.86 \begin {gather*} \frac {\ln \left (x^8+x^4+1\right )}{8}-\frac {\sqrt {3}\,\mathrm {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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