Integrand size = 13, antiderivative size = 23 \[ \int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx=\frac {\sqrt [4]{-1+x^4} \left (1+4 x^4\right )}{5 x^5} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {277, 270} \[ \int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{x^4-1}}{5 x}+\frac {\sqrt [4]{x^4-1}}{5 x^5} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-1+x^4}}{5 x^5}+\frac {4}{5} \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx \\ & = \frac {\sqrt [4]{-1+x^4}}{5 x^5}+\frac {4 \sqrt [4]{-1+x^4}}{5 x} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx=\frac {\sqrt [4]{-1+x^4} \left (1+4 x^4\right )}{5 x^5} \]
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Time = 0.85 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
trager | \(\frac {\left (x^{4}-1\right )^{\frac {1}{4}} \left (4 x^{4}+1\right )}{5 x^{5}}\) | \(20\) |
pseudoelliptic | \(\frac {\left (x^{4}-1\right )^{\frac {1}{4}} \left (4 x^{4}+1\right )}{5 x^{5}}\) | \(20\) |
risch | \(\frac {4 x^{8}-3 x^{4}-1}{5 \left (x^{4}-1\right )^{\frac {3}{4}} x^{5}}\) | \(25\) |
gosper | \(\frac {\left (x^{2}+1\right ) \left (x -1\right ) \left (1+x \right ) \left (4 x^{4}+1\right )}{5 x^{5} \left (x^{4}-1\right )^{\frac {3}{4}}}\) | \(31\) |
meijerg | \(-\frac {{\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {3}{4}} \left (4 x^{4}+1\right ) \left (-x^{4}+1\right )^{\frac {1}{4}}}{5 \operatorname {signum}\left (x^{4}-1\right )^{\frac {3}{4}} x^{5}}\) | \(40\) |
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none
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx=\frac {{\left (4 \, x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \]
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Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 124, normalized size of antiderivative = 5.39 \[ \int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx=\begin {cases} - \frac {\sqrt [4]{-1 + \frac {1}{x^{4}}} e^{- \frac {3 i \pi }{4}} \Gamma \left (- \frac {5}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} - \frac {\sqrt [4]{-1 + \frac {1}{x^{4}}} e^{- \frac {3 i \pi }{4}} \Gamma \left (- \frac {5}{4}\right )}{16 x^{4} \Gamma \left (\frac {3}{4}\right )} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\\frac {\sqrt [4]{1 - \frac {1}{x^{4}}} \Gamma \left (- \frac {5}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} + \frac {\sqrt [4]{1 - \frac {1}{x^{4}}} \Gamma \left (- \frac {5}{4}\right )}{16 x^{4} \Gamma \left (\frac {3}{4}\right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx=\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} - \frac {{\left (x^{4} - 1\right )}^{\frac {5}{4}}}{5 \, x^{5}} \]
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\[ \int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (x^{4} - 1\right )}^{\frac {3}{4}} x^{6}} \,d x } \]
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Time = 5.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx=\frac {{\left (x^4-1\right )}^{1/4}+4\,x^4\,{\left (x^4-1\right )}^{1/4}}{5\,x^5} \]
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