Optimal. Leaf size=38 \[ \frac {3 \left (x^4-1\right )^{2/3} \left (5 x^8-4 x^7-10 x^4+4 x^3+5\right )}{20 x^8} \]
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Rubi [A] time = 0.09, antiderivative size = 33, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1833, 1584, 449, 1474, 847, 74} \begin {gather*} \frac {3 \left (x^4-1\right )^{8/3}}{4 x^8}-\frac {3 \left (x^4-1\right )^{5/3}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rule 449
Rule 847
Rule 1474
Rule 1584
Rule 1833
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^9} \, dx &=\int \left (\frac {\left (-1+x^4\right )^{2/3} \left (-3 x^2-x^6\right )}{x^8}+\frac {\left (-1+x^4\right )^{2/3} \left (-6+4 x^4+2 x^8\right )}{x^9}\right ) \, dx\\ &=\int \frac {\left (-1+x^4\right )^{2/3} \left (-3 x^2-x^6\right )}{x^8} \, dx+\int \frac {\left (-1+x^4\right )^{2/3} \left (-6+4 x^4+2 x^8\right )}{x^9} \, dx\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {(-1+x)^{2/3} \left (-6+4 x+2 x^2\right )}{x^3} \, dx,x,x^4\right )+\int \frac {\left (-3-x^4\right ) \left (-1+x^4\right )^{2/3}}{x^6} \, dx\\ &=-\frac {3 \left (-1+x^4\right )^{5/3}}{5 x^5}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {(-1+x)^{5/3} (6+2 x)}{x^3} \, dx,x,x^4\right )\\ &=-\frac {3 \left (-1+x^4\right )^{5/3}}{5 x^5}+\frac {3 \left (-1+x^4\right )^{8/3}}{4 x^8}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 155, normalized size = 4.08 \begin {gather*} \frac {\left (x^4-1\right )^{2/3} \left (x^4 \left (20 \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )+3 x \left (1-x^4\right )^{2/3} \left (-6 x^4 \, _2F_1\left (\frac {5}{3},3;\frac {8}{3};1-x^4\right )-5 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};1-x^4\right )+4 \left (x^4-1\right ) \, _2F_1\left (\frac {5}{3},2;\frac {8}{3};1-x^4\right )+6 \, _2F_1\left (\frac {5}{3},3;\frac {8}{3};1-x^4\right )+5\right )\right )+12 \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )\right )}{20 x^5 \left (1-x^4\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 28, normalized size = 0.74 \begin {gather*} \frac {3 \left (-1+x^4\right )^{5/3} \left (-5-4 x^3+5 x^4\right )}{20 x^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 34, normalized size = 0.89 \begin {gather*} \frac {3 \, {\left (5 \, x^{8} - 4 \, x^{7} - 10 \, x^{4} + 4 \, x^{3} + 5\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{20 \, x^{8}} \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 (2 \, x^{4} - x^{3} - 2\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 35, normalized size = 0.92
method | result | size |
trager | \(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}} \left (5 x^{8}-4 x^{7}-10 x^{4}+4 x^{3}+5\right )}{20 x^{8}}\) | \(35\) |
gosper | \(\frac {3 \left (x^{2}+1\right ) \left (-1+x \right ) \left (1+x \right ) \left (5 x^{4}-4 x^{3}-5\right ) \left (x^{4}-1\right )^{\frac {2}{3}}}{20 x^{8}}\) | \(36\) |
risch | \(\frac {-\frac {3}{5} x^{11}+\frac {6}{5} x^{7}-\frac {3}{5} x^{3}-\frac {9}{4} x^{8}+\frac {9}{4} x^{4}-\frac {3}{4}+\frac {3}{4} x^{12}}{x^{8} \left (x^{4}-1\right )^{\frac {1}{3}}}\) | \(45\) |
meijerg | \(-\frac {\mathrm {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right ) \sqrt {3}\, \left (-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}+\frac {2 \hypergeom \left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], x^{4}\right ) \pi \sqrt {3}\, x^{4}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{6 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {2}{3}} \pi }+\frac {\mathrm {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {2}{3}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], x^{4}\right )}{\left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {2}{3}} x}+\frac {\mathrm {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right ) \sqrt {3}\, \left (-\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{4}}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+4 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], x^{4}\right ) \pi \sqrt {3}\, x^{4}}{9 \Gamma \left (\frac {2}{3}\right )}\right )}{3 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {2}{3}} \pi }+\frac {3 \mathrm {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {5}{4}, -\frac {2}{3}\right ], \left [-\frac {1}{4}\right ], x^{4}\right )}{5 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {2}{3}} x^{5}}+\frac {\mathrm {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right ) \sqrt {3}\, \left (\frac {\pi \sqrt {3}}{2 \Gamma \left (\frac {2}{3}\right ) x^{8}}-\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{4}}+\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {4 \hypergeom \left (\left [1, 1, \frac {7}{3}\right ], \left [2, 4\right ], x^{4}\right ) \pi \sqrt {3}\, x^{4}}{81 \Gamma \left (\frac {2}{3}\right )}\right )}{2 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {2}{3}} \pi }\) | \(353\) |
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*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {2 \, {\left (x^{4} - 1\right )}^{\frac {5}{3}} - {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{4 \, {\left (2 \, x^{4} + {\left (x^{4} - 1\right )}^{2} - 1\right )}} + \int \frac {{\left (2 \, x^{5} - x^{4} + 4 \, x - 3\right )} {\left (x^{2} + 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {2}{3}}}{x^{6}}\,{d x} - \frac {1}{12} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 58, normalized size = 1.53 \begin {gather*} \frac {3\,{\left (x^4-1\right )}^{2/3}}{4}-\frac {3\,{\left (x^4-1\right )}^{2/3}}{5\,x}-\frac {3\,{\left (x^4-1\right )}^{2/3}}{2\,x^4}+\frac {3\,{\left (x^4-1\right )}^{2/3}}{5\,x^5}+\frac {3\,{\left (x^4-1\right )}^{2/3}}{4\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.95, size = 187, normalized size = 4.92 \begin {gather*} - \frac {x^{\frac {8}{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{2 \Gamma \left (\frac {1}{3}\right )} + \frac {e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} + \frac {3 e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {2}{3} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} - \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{x^{\frac {4}{3}} \Gamma \left (\frac {4}{3}\right )} + \frac {3 \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{2 x^{\frac {16}{3}} \Gamma \left (\frac {7}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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