Integrand size = 20, antiderivative size = 25 \[ \int \frac {-1+x^4}{x^8 \sqrt [4]{-1+2 x^4}} \, dx=\frac {\left (-3-x^4\right ) \left (-1+2 x^4\right )^{3/4}}{21 x^7} \]
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Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {464, 270} \[ \int \frac {-1+x^4}{x^8 \sqrt [4]{-1+2 x^4}} \, dx=-\frac {\left (2 x^4-1\right )^{3/4}}{7 x^7}-\frac {\left (2 x^4-1\right )^{3/4}}{21 x^3} \]
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Rule 270
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (-1+2 x^4\right )^{3/4}}{7 x^7}-\frac {1}{7} \int \frac {1}{x^4 \sqrt [4]{-1+2 x^4}} \, dx \\ & = -\frac {\left (-1+2 x^4\right )^{3/4}}{7 x^7}-\frac {\left (-1+2 x^4\right )^{3/4}}{21 x^3} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^4}{x^8 \sqrt [4]{-1+2 x^4}} \, dx=\frac {\left (-3-x^4\right ) \left (-1+2 x^4\right )^{3/4}}{21 x^7} \]
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Time = 0.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80
method | result | size |
gosper | \(-\frac {\left (x^{4}+3\right ) \left (2 x^{4}-1\right )^{\frac {3}{4}}}{21 x^{7}}\) | \(20\) |
trager | \(-\frac {\left (x^{4}+3\right ) \left (2 x^{4}-1\right )^{\frac {3}{4}}}{21 x^{7}}\) | \(20\) |
pseudoelliptic | \(-\frac {\left (x^{4}+3\right ) \left (2 x^{4}-1\right )^{\frac {3}{4}}}{21 x^{7}}\) | \(20\) |
risch | \(-\frac {2 x^{8}+5 x^{4}-3}{21 x^{7} \left (2 x^{4}-1\right )^{\frac {1}{4}}}\) | \(27\) |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (2 x^{4}-1\right )\right )}^{\frac {1}{4}} \left (1+\frac {8 x^{4}}{3}\right ) \left (-2 x^{4}+1\right )^{\frac {3}{4}}}{7 \operatorname {signum}\left (2 x^{4}-1\right )^{\frac {1}{4}} x^{7}}-\frac {{\left (-\operatorname {signum}\left (2 x^{4}-1\right )\right )}^{\frac {1}{4}} \left (-2 x^{4}+1\right )^{\frac {3}{4}}}{3 \operatorname {signum}\left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}}\) | \(81\) |
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {-1+x^4}{x^8 \sqrt [4]{-1+2 x^4}} \, dx=-\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} + 3\right )}}{21 \, x^{7}} \]
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Result contains complex when optimal does not.
Time = 1.06 (sec) , antiderivative size = 230, normalized size of antiderivative = 9.20 \[ \int \frac {-1+x^4}{x^8 \sqrt [4]{-1+2 x^4}} \, dx=\begin {cases} - \frac {2^{\frac {3}{4}} \left (-1 + \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {3}{4}\right )}{4 \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 2 \\- \frac {2^{\frac {3}{4}} \left (1 - \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{4 \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} - \begin {cases} - \frac {2^{\frac {3}{4}} \left (-1 + \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{4}\right )}{2 \Gamma \left (\frac {1}{4}\right )} - \frac {3 \cdot 2^{\frac {3}{4}} \left (-1 + \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{4}\right )}{16 x^{4} \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 2 \\\frac {2^{\frac {3}{4}} \left (1 - \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{2 \Gamma \left (\frac {1}{4}\right )} + \frac {3 \cdot 2^{\frac {3}{4}} \left (1 - \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{16 x^{4} \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-1+x^4}{x^8 \sqrt [4]{-1+2 x^4}} \, dx=-\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} + \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} \]
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\[ \int \frac {-1+x^4}{x^8 \sqrt [4]{-1+2 x^4}} \, dx=\int { \frac {x^{4} - 1}{{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{8}} \,d x } \]
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Time = 5.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {-1+x^4}{x^8 \sqrt [4]{-1+2 x^4}} \, dx=-\frac {x^4\,{\left (2\,x^4-1\right )}^{3/4}+3\,{\left (2\,x^4-1\right )}^{3/4}}{21\,x^7} \]
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