Integrand size = 22, antiderivative size = 20 \[ \int \frac {1+x^4}{x^2 \sqrt [4]{-x^2+x^6}} \, dx=\frac {2 \left (-x^2+x^6\right )^{3/4}}{3 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1604} \[ \int \frac {1+x^4}{x^2 \sqrt [4]{-x^2+x^6}} \, dx=\frac {2 \left (x^6-x^2\right )^{3/4}}{3 x^3} \]
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Rule 1604
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (-x^2+x^6\right )^{3/4}}{3 x^3} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^4}{x^2 \sqrt [4]{-x^2+x^6}} \, dx=\frac {2 \left (x^2 \left (-1+x^4\right )\right )^{3/4}}{3 x^3} \]
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Time = 0.86 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
trager | \(\frac {2 \left (x^{6}-x^{2}\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(17\) |
pseudoelliptic | \(\frac {2 \left (x^{6}-x^{2}\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(17\) |
risch | \(\frac {\frac {2 x^{4}}{3}-\frac {2}{3}}{x \left (x^{2} \left (x^{4}-1\right )\right )^{\frac {1}{4}}}\) | \(22\) |
gosper | \(\frac {2 \left (x^{2}+1\right ) \left (x -1\right ) \left (1+x \right )}{3 x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}\) | \(28\) |
meijerg | \(-\frac {2 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {3}{8}, \frac {1}{4}\right ], \left [\frac {5}{8}\right ], x^{4}\right )}{3 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {3}{2}}}+\frac {2 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} x^{\frac {5}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {5}{8}\right ], \left [\frac {13}{8}\right ], x^{4}\right )}{5 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\) | \(66\) |
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none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1+x^4}{x^2 \sqrt [4]{-x^2+x^6}} \, dx=\frac {2 \, {\left (x^{6} - x^{2}\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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\[ \int \frac {1+x^4}{x^2 \sqrt [4]{-x^2+x^6}} \, dx=\int \frac {x^{4} + 1}{x^{2} \sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {1+x^4}{x^2 \sqrt [4]{-x^2+x^6}} \, dx=\int { \frac {x^{4} + 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} x^{2}} \,d x } \]
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\[ \int \frac {1+x^4}{x^2 \sqrt [4]{-x^2+x^6}} \, dx=\int { \frac {x^{4} + 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} x^{2}} \,d x } \]
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Time = 5.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1+x^4}{x^2 \sqrt [4]{-x^2+x^6}} \, dx=\frac {2\,{\left (x^6-x^2\right )}^{3/4}}{3\,x^3} \]
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