Integrand size = 16, antiderivative size = 39 \[ \int \frac {x^7}{1-x^4+x^8} \, dx=-\frac {\arctan \left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{8} \log \left (1-x^4+x^8\right ) \]
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Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1371, 648, 632, 210, 642} \[ \int \frac {x^7}{1-x^4+x^8} \, dx=\frac {1}{8} \log \left (x^8-x^4+1\right )-\frac {\arctan \left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {x}{1-x+x^2} \, dx,x,x^4\right ) \\ & = \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 ) \\ & = \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*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {x^7}{1-x^4+x^8} \, dx=\frac {\arctan \left (\frac {-1+2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{8} \log \left (1-x^4+x^8\right ) \]
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Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {\ln \left (x^{8}-x^{4}+1\right )}{8}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{4}-1\right ) \sqrt {3}}{3}\right )}{12}\) | \(33\) |
risch | \(\frac {\ln \left (4 x^{8}-4 x^{4}+4\right )}{8}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{4}-1\right ) \sqrt {3}}{3}\right )}{12}\) | \(35\) |
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Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {x^7}{1-x^4+x^8} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) \]
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Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {x^7}{1-x^4+x^8} \, dx=\frac {\log {\left (x^{8} - x^{4} + 1 \right )}}{8} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} - \frac {\sqrt {3}}{3} \right )}}{12} \]
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Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {x^7}{1-x^4+x^8} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {x^7}{1-x^4+x^8} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) \]
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Time = 8.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {x^7}{1-x^4+x^8} \, dx=\frac {\ln \left (x^8-x^4+1\right )}{8}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}-\frac {2\,\sqrt {3}\,x^4}{3}\right )}{12} \]
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