Optimal. Leaf size=20 \[ \frac {2 \left (x^6-x^2\right )^{7/4}}{7 x^7} \]
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Rubi [B] time = 0.16, antiderivative size = 41, normalized size of antiderivative = 2.05, number of steps used = 12, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2052, 2025, 2011, 329, 246, 245, 2024} \begin {gather*} \frac {2 \left (x^6-x^2\right )^{3/4}}{7 x}-\frac {2 \left (x^6-x^2\right )^{3/4}}{7 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 329
Rule 2011
Rule 2024
Rule 2025
Rule 2052
Rubi steps
\begin {align*} \int \frac {-1+x^8}{x^4 \sqrt [4]{-x^2+x^6}} \, dx &=\int \left (-\frac {1}{x^4 \sqrt [4]{-x^2+x^6}}+\frac {x^4}{\sqrt [4]{-x^2+x^6}}\right ) \, dx\\ &=-\int \frac {1}{x^4 \sqrt [4]{-x^2+x^6}} \, dx+\int \frac {x^4}{\sqrt [4]{-x^2+x^6}} \, dx\\ &=-\frac {2 \left (-x^2+x^6\right )^{3/4}}{7 x^5}+\frac {2 \left (-x^2+x^6\right )^{3/4}}{7 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^2 \left (x^4-1\right )\right )^{7/4}}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \left (-x^2+x^6\right )^{7/4}}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 21, normalized size = 1.05 \begin {gather*} \frac {2 \, {\left (x^{6} - x^{2}\right )}^{\frac {3}{4}} {\left (x^{4} - 1\right )}}{7 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} - 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 22, normalized size = 1.10
method | result | size |
trager | \(\frac {2 \left (x^{4}-1\right ) \left (x^{6}-x^{2}\right )^{\frac {3}{4}}}{7 x^{5}}\) | \(22\) |
risch | \(\frac {\frac {2}{7} x^{8}-\frac {4}{7} x^{4}+\frac {2}{7}}{x^{3} \left (x^{2} \left (x^{4}-1\right )\right )^{\frac {1}{4}}}\) | \(27\) |
gosper | \(\frac {2 \left (x^{4}-1\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}{7 \left (x^{6}-x^{2}\right )^{\frac {1}{4}} x^{3}}\) | \(33\) |
meijerg | \(\frac {2 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {7}{8}, \frac {1}{4}\right ], \left [\frac {1}{8}\right ], x^{4}\right )}{7 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {7}{2}}}+\frac {2 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {1}{4}, \frac {9}{8}\right ], \left [\frac {17}{8}\right ], x^{4}\right ) x^{\frac {9}{2}}}{9 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} - 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 33, normalized size = 1.65 \begin {gather*} \frac {2\,{\left (x^6-x^2\right )}^{3/4}}{7\,x}-\frac {2\,{\left (x^6-x^2\right )}^{3/4}}{7\,x^5} \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 {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{x^{4} \sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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