Optimal. Leaf size=37 \[ \frac {2 \left (x^4-x^2\right )^{3/4} \left (128 x^6+96 x^4+84 x^2+77\right )}{1155 x^9} \]
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Rubi [B] time = 0.11, antiderivative size = 81, normalized size of antiderivative = 2.19, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2016, 2014} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^8 \sqrt [4]{-x^2+x^4}} \, dx &=\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*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^2 \left (x^2-1\right )\right )^{3/4} \left (128 x^6+96 x^4+84 x^2+77\right )}{1155 x^9} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 37, normalized size = 1.00 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 33, normalized size = 0.89 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 59, normalized size = 1.59 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 34, normalized size = 0.92
method | result | size |
trager | \(\frac {2 \left (x^{4}-x^{2}\right )^{\frac {3}{4}} \left (128 x^{6}+96 x^{4}+84 x^{2}+77\right )}{1155 x^{9}}\) | \(34\) |
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 | \(\frac {2 \left (-1+x \right ) \left (1+x \right ) \left (128 x^{6}+96 x^{4}+84 x^{2}+77\right )}{1155 x^{7} \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\) | \(40\) |
meijerg | \(-\frac {2 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{4}} \left (\frac {128}{77} x^{6}+\frac {96}{77} x^{4}+\frac {12}{11} x^{2}+1\right ) \left (-x^{2}+1\right )^{\frac {3}{4}}}{15 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{4}} x^{\frac {15}{2}}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 39, normalized size = 1.05 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 65, normalized size = 1.76 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{8} \sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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