Integrand size = 17, antiderivative size = 37 \[ \int \frac {1}{x^8 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \left (-x^2+x^4\right )^{3/4} \left (77+84 x^2+96 x^4+128 x^6\right )}{1155 x^9} \]
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Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(37)=74\).
Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.19, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2041, 2039} \[ \int \frac {1}{x^8 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \left (x^4-x^2\right )^{3/4}}{15 x^9}+\frac {8 \left (x^4-x^2\right )^{3/4}}{55 x^7}+\frac {64 \left (x^4-x^2\right )^{3/4}}{385 x^5}+\frac {256 \left (x^4-x^2\right )^{3/4}}{1155 x^3} \]
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Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (-x^2+x^4\right )^{3/4}}{15 x^9}+\frac {4}{5} \int \frac {1}{x^6 \sqrt [4]{-x^2+x^4}} \, dx \\ & = \frac {2 \left (-x^2+x^4\right )^{3/4}}{15 x^9}+\frac {8 \left (-x^2+x^4\right )^{3/4}}{55 x^7}+\frac {32}{55} \int \frac {1}{x^4 \sqrt [4]{-x^2+x^4}} \, dx \\ & = \frac {2 \left (-x^2+x^4\right )^{3/4}}{15 x^9}+\frac {8 \left (-x^2+x^4\right )^{3/4}}{55 x^7}+\frac {64 \left (-x^2+x^4\right )^{3/4}}{385 x^5}+\frac {128}{385} \int \frac {1}{x^2 \sqrt [4]{-x^2+x^4}} \, dx \\ & = \frac {2 \left (-x^2+x^4\right )^{3/4}}{15 x^9}+\frac {8 \left (-x^2+x^4\right )^{3/4}}{55 x^7}+\frac {64 \left (-x^2+x^4\right )^{3/4}}{385 x^5}+\frac {256 \left (-x^2+x^4\right )^{3/4}}{1155 x^3} \\ \end{align*}
Time = 1.34 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^8 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \left (x^2 \left (-1+x^2\right )\right )^{3/4} \left (77+84 x^2+96 x^4+128 x^6\right )}{1155 x^9} \]
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Time = 0.87 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92
method | result | size |
trager | \(\frac {2 \left (x^{4}-x^{2}\right )^{\frac {3}{4}} \left (128 x^{6}+96 x^{4}+84 x^{2}+77\right )}{1155 x^{9}}\) | \(34\) |
pseudoelliptic | \(\frac {2 \left (x^{4}-x^{2}\right )^{\frac {3}{4}} \left (128 x^{6}+96 x^{4}+84 x^{2}+77\right )}{1155 x^{9}}\) | \(34\) |
risch | \(\frac {-\frac {2}{165} x^{2}-\frac {2}{15}-\frac {8}{385} x^{4}-\frac {64}{1155} x^{6}+\frac {256}{1155} x^{8}}{x^{7} \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}}}\) | \(39\) |
gosper | \(\frac {2 \left (1+x \right ) \left (x -1\right ) \left (128 x^{6}+96 x^{4}+84 x^{2}+77\right )}{1155 x^{7} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\) | \(40\) |
meijerg | \(-\frac {2 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{4}} \left (\frac {128}{77} x^{6}+\frac {96}{77} x^{4}+\frac {12}{11} x^{2}+1\right ) \left (-x^{2}+1\right )^{\frac {3}{4}}}{15 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{4}} x^{\frac {15}{2}}}\) | \(50\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^8 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \, {\left (128 \, x^{6} + 96 \, x^{4} + 84 \, x^{2} + 77\right )} {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{1155 \, x^{9}} \]
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\[ \int \frac {1}{x^8 \sqrt [4]{-x^2+x^4}} \, dx=\int \frac {1}{x^{8} \sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^8 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \, {\left (128 \, x^{9} - 32 \, x^{7} - 12 \, x^{5} - 7 \, x^{3} - 77 \, x\right )}}{1155 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {17}{2}}} \]
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Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.59 \[ \int \frac {1}{x^8 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2}{15} \, {\left (\frac {1}{x^{2}} - 1\right )}^{3} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}} + \frac {6}{11} \, {\left (\frac {1}{x^{2}} - 1\right )}^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}} - \frac {6}{7} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {7}{4}} + \frac {2}{3} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}} \]
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Time = 4.99 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.76 \[ \int \frac {1}{x^8 \sqrt [4]{-x^2+x^4}} \, dx=\frac {256\,{\left (x^4-x^2\right )}^{3/4}}{1155\,x^3}+\frac {64\,{\left (x^4-x^2\right )}^{3/4}}{385\,x^5}+\frac {8\,{\left (x^4-x^2\right )}^{3/4}}{55\,x^7}+\frac {2\,{\left (x^4-x^2\right )}^{3/4}}{15\,x^9} \]
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