Integrand size = 24, antiderivative size = 20 \[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=\frac {x \left (1-x^8\right )^{3/4}}{-1+x^8} \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {21, 391} \[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=-\frac {x}{\sqrt [4]{1-x^8}} \]
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Rule 21
Rule 391
Rubi steps \begin{align*} \text {integral}& = -\int \frac {1+x^8}{\left (1-x^8\right )^{5/4}} \, dx \\ & = -\frac {x}{\sqrt [4]{1-x^8}} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=-\frac {x}{\sqrt [4]{1-x^8}} \]
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Time = 1.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.65
method | result | size |
gosper | \(-\frac {x}{\left (-x^{8}+1\right )^{\frac {1}{4}}}\) | \(13\) |
risch | \(-\frac {x}{\left (-x^{8}+1\right )^{\frac {1}{4}}}\) | \(13\) |
pseudoelliptic | \(-\frac {x}{\left (-x^{8}+1\right )^{\frac {1}{4}}}\) | \(13\) |
trager | \(\frac {x \left (-x^{8}+1\right )^{\frac {3}{4}}}{x^{8}-1}\) | \(19\) |
meijerg | \(-x \operatorname {hypergeom}\left (\left [\frac {1}{8}, \frac {5}{4}\right ], \left [\frac {9}{8}\right ], x^{8}\right )-\frac {x^{9} \operatorname {hypergeom}\left (\left [\frac {9}{8}, \frac {5}{4}\right ], \left [\frac {17}{8}\right ], x^{8}\right )}{9}\) | \(28\) |
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none
Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=\frac {{\left (-x^{8} + 1\right )}^{\frac {3}{4}} x}{x^{8} - 1} \]
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\[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=\int \frac {x^{8} + 1}{\sqrt [4]{- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \]
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none
Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=-\frac {x}{{\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \]
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\[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=\int { \frac {x^{8} + 1}{{\left (x^{8} - 1\right )} {\left (-x^{8} + 1\right )}^{\frac {1}{4}}} \,d x } \]
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Time = 5.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.60 \[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=-\frac {x}{{\left (1-x^8\right )}^{1/4}} \]
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