Integrand size = 16, antiderivative size = 23 \[ \int \frac {x^3}{1+3 x^4+x^8} \, dx=-\frac {\text {arctanh}\left (\frac {3+2 x^4}{\sqrt {5}}\right )}{2 \sqrt {5}} \]
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Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1366, 632, 212} \[ \int \frac {x^3}{1+3 x^4+x^8} \, dx=-\frac {\text {arctanh}\left (\frac {2 x^4+3}{\sqrt {5}}\right )}{2 \sqrt {5}} \]
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Rule 212
Rule 632
Rule 1366
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {1}{1+3 x+x^2} \, dx,x,x^4\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{5-x^2} \, dx,x,3+2 x^4\right )\right ) \\ & = -\frac {\tanh ^{-1}\left (\frac {3+2 x^4}{\sqrt {5}}\right )}{2 \sqrt {5}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {x^3}{1+3 x^4+x^8} \, dx=\frac {\log \left (-3+\sqrt {5}-2 x^4\right )-\log \left (3+\sqrt {5}+2 x^4\right )}{4 \sqrt {5}} \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {\operatorname {arctanh}\left (\frac {\left (2 x^{4}+3\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{10}\) | \(19\) |
risch | \(\frac {\ln \left (2 x^{4}-\sqrt {5}+3\right ) \sqrt {5}}{20}-\frac {\ln \left (2 x^{4}+\sqrt {5}+3\right ) \sqrt {5}}{20}\) | \(36\) |
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (18) = 36\).
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int \frac {x^3}{1+3 x^4+x^8} \, dx=\frac {1}{20} \, \sqrt {5} \log \left (\frac {2 \, x^{8} + 6 \, x^{4} - \sqrt {5} {\left (2 \, x^{4} + 3\right )} + 7}{x^{8} + 3 \, x^{4} + 1}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {x^3}{1+3 x^4+x^8} \, dx=\frac {\sqrt {5} \log {\left (x^{4} - \frac {\sqrt {5}}{2} + \frac {3}{2} \right )}}{20} - \frac {\sqrt {5} \log {\left (x^{4} + \frac {\sqrt {5}}{2} + \frac {3}{2} \right )}}{20} \]
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none
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {x^3}{1+3 x^4+x^8} \, dx=\frac {1}{20} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - \sqrt {5} + 3}{2 \, x^{4} + \sqrt {5} + 3}\right ) \]
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none
Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {x^3}{1+3 x^4+x^8} \, dx=\frac {1}{20} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - \sqrt {5} + 3}{2 \, x^{4} + \sqrt {5} + 3}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {x^3}{1+3 x^4+x^8} \, dx=\frac {\sqrt {5}\,\mathrm {atanh}\left (\frac {8\,\sqrt {5}\,x^4+3\,\sqrt {5}}{18\,x^4+7}\right )}{10} \]
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