Optimal. Leaf size=28 \[ \frac {\left (x^4-1\right )^{3/4} \left (-5 x^8-23 x^4+28\right )}{77 x^{11}} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {453, 264} \begin {gather*} -\frac {4 \left (x^4-1\right )^{7/4}}{11 x^{11}}-\frac {5 \left (x^4-1\right )^{7/4}}{77 x^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {\left (-4+x^4\right ) \left (-1+x^4\right )^{3/4}}{x^{12}} \, dx &=-\frac {4 \left (-1+x^4\right )^{7/4}}{11 x^{11}}-\frac {5}{11} \int \frac {\left (-1+x^4\right )^{3/4}}{x^8} \, dx\\ &=-\frac {4 \left (-1+x^4\right )^{7/4}}{11 x^{11}}-\frac {5 \left (-1+x^4\right )^{7/4}}{77 x^7}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.82 \begin {gather*} -\frac {\left (x^4-1\right )^{7/4} \left (5 x^4+28\right )}{77 x^{11}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 23, normalized size = 0.82 \begin {gather*} \frac {\left (-28-5 x^4\right ) \left (-1+x^4\right )^{7/4}}{77 x^{11}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 24, normalized size = 0.86 \begin {gather*} -\frac {{\left (5 \, x^{8} + 23 \, x^{4} - 28\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \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 {{\left (x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} - 4\right )}}{x^{12}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 25, normalized size = 0.89
method | result | size |
trager | \(-\frac {\left (5 x^{8}+23 x^{4}-28\right ) \left (x^{4}-1\right )^{\frac {3}{4}}}{77 x^{11}}\) | \(25\) |
risch | \(-\frac {5 x^{12}+18 x^{8}-51 x^{4}+28}{77 x^{11} \left (x^{4}-1\right )^{\frac {1}{4}}}\) | \(30\) |
gosper | \(-\frac {\left (x^{2}+1\right ) \left (1+x \right ) \left (-1+x \right ) \left (5 x^{4}+28\right ) \left (x^{4}-1\right )^{\frac {3}{4}}}{77 x^{11}}\) | \(31\) |
meijerg | \(\frac {4 \mathrm {signum}\left (x^{4}-1\right )^{\frac {3}{4}} \left (-\frac {4}{7} x^{8}-\frac {3}{7} x^{4}+1\right ) \left (-x^{4}+1\right )^{\frac {3}{4}}}{11 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {3}{4}} x^{11}}-\frac {\mathrm {signum}\left (x^{4}-1\right )^{\frac {3}{4}} \left (-x^{4}+1\right )^{\frac {7}{4}}}{7 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {3}{4}} x^{7}}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 25, normalized size = 0.89 \begin {gather*} -\frac {3 \, {\left (x^{4} - 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} + \frac {4 \, {\left (x^{4} - 1\right )}^{\frac {11}{4}}}{11 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 39, normalized size = 1.39 \begin {gather*} -\frac {23\,x^4\,{\left (x^4-1\right )}^{3/4}-28\,{\left (x^4-1\right )}^{3/4}+5\,x^8\,{\left (x^4-1\right )}^{3/4}}{77\,x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.61, size = 420, normalized size = 15.00 \begin {gather*} \begin {cases} \frac {\left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{4}\right )}{4 \Gamma \left (- \frac {3}{4}\right )} - \frac {\left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{4}\right )}{4 x^{4} \Gamma \left (- \frac {3}{4}\right )} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\- \frac {\left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{4 \Gamma \left (- \frac {3}{4}\right )} + \frac {\left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{4 x^{4} \Gamma \left (- \frac {3}{4}\right )} & \text {otherwise} \end {cases} - 4 \left (\begin {cases} \frac {\left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {11}{4}\right )}{4 \Gamma \left (- \frac {3}{4}\right )} + \frac {3 \left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{4} \Gamma \left (- \frac {3}{4}\right )} - \frac {7 \left (-1 + \frac {1}{x^{4}}\right )^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{8} \Gamma \left (- \frac {3}{4}\right )} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\\frac {4 x^{8} \left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{8} \Gamma \left (- \frac {3}{4}\right ) - 16 x^{4} \Gamma \left (- \frac {3}{4}\right )} - \frac {x^{4} \left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{8} \Gamma \left (- \frac {3}{4}\right ) - 16 x^{4} \Gamma \left (- \frac {3}{4}\right )} + \frac {7 \left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{12} \Gamma \left (- \frac {3}{4}\right ) - 16 x^{8} \Gamma \left (- \frac {3}{4}\right )} - \frac {10 \left (1 - \frac {1}{x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {11}{4}\right )}{16 x^{8} \Gamma \left (- \frac {3}{4}\right ) - 16 x^{4} \Gamma \left (- \frac {3}{4}\right )} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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