Optimal. Leaf size=28 \[ \frac {2 \left (x^6+1\right )^{3/4} \left (3 x^6-7 x^4+3\right )}{21 x^7} \]
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Rubi [A] time = 0.08, antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1833, 1584, 449, 1478} \begin {gather*} \frac {2 \left (x^6+1\right )^{7/4}}{7 x^7}-\frac {2 \left (x^6+1\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 1478
Rule 1584
Rule 1833
Rubi steps
\begin {align*} \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^8 \sqrt [4]{1+x^6}} \, dx &=\int \left (\frac {2 x^3-x^9}{x^7 \sqrt [4]{1+x^6}}+\frac {-2-x^6+x^{12}}{x^8 \sqrt [4]{1+x^6}}\right ) \, dx\\ &=\int \frac {2 x^3-x^9}{x^7 \sqrt [4]{1+x^6}} \, dx+\int \frac {-2-x^6+x^{12}}{x^8 \sqrt [4]{1+x^6}} \, dx\\ &=\int \frac {2-x^6}{x^4 \sqrt [4]{1+x^6}} \, dx+\int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4}}{x^8} \, dx\\ &=-\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}+\frac {2 \left (1+x^6\right )^{7/4}}{7 x^7}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 108, normalized size = 3.86 \begin {gather*} \frac {\, _2F_1\left (-\frac {1}{6},\frac {1}{4};\frac {5}{6};-x^6\right )}{x}+\frac {2 \, _2F_1\left (-\frac {7}{6},\frac {1}{4};-\frac {1}{6};-x^6\right )}{7 x^7}+\frac {1}{5} x^5 \, _2F_1\left (\frac {1}{4},\frac {5}{6};\frac {11}{6};-x^6\right )-\frac {1}{3} x^3 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-x^6\right )-\frac {2 \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};-x^6\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.09, size = 28, normalized size = 1.00 \begin {gather*} \frac {2 \left (1+x^6\right )^{3/4} \left (3-7 x^4+3 x^6\right )}{21 x^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 24, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (3 \, x^{6} - 7 \, x^{4} + 3\right )} {\left (x^{6} + 1\right )}^{\frac {3}{4}}}{21 \, x^{7}} \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^{6} - x^{4} + 1\right )} {\left (x^{6} - 2\right )}}{{\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{8}}\,{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 | \(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}} \left (3 x^{6}-7 x^{4}+3\right )}{21 x^{7}}\) | \(25\) |
risch | \(\frac {\frac {2}{7} x^{12}+\frac {4}{7} x^{6}-\frac {2}{3} x^{10}-\frac {2}{3} x^{4}+\frac {2}{7}}{x^{7} \left (x^{6}+1\right )^{\frac {1}{4}}}\) | \(35\) |
gosper | \(\frac {2 \left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right ) \left (3 x^{6}-7 x^{4}+3\right )}{21 x^{7} \left (x^{6}+1\right )^{\frac {1}{4}}}\) | \(40\) |
meijerg | \(\frac {2 \hypergeom \left (\left [-\frac {7}{6}, \frac {1}{4}\right ], \left [-\frac {1}{6}\right ], -x^{6}\right )}{7 x^{7}}+\frac {\hypergeom \left (\left [-\frac {1}{6}, \frac {1}{4}\right ], \left [\frac {5}{6}\right ], -x^{6}\right )}{x}-\frac {2 \hypergeom \left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {1}{2}\right ], -x^{6}\right )}{3 x^{3}}+\frac {\hypergeom \left (\left [\frac {1}{4}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], -x^{6}\right ) x^{5}}{5}-\frac {\hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{6}\right ) x^{3}}{3}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 46, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left (3 \, x^{12} - 7 \, x^{10} + 6 \, x^{6} - 7 \, x^{4} + 3\right )}}{21 \, {\left (x^{4} - x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}^{\frac {1}{4}} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 39, normalized size = 1.39 \begin {gather*} \frac {6\,{\left (x^6+1\right )}^{3/4}-14\,x^4\,{\left (x^6+1\right )}^{3/4}+6\,x^6\,{\left (x^6+1\right )}^{3/4}}{21\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.61, size = 143, normalized size = 5.11 \begin {gather*} \frac {x^{5} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} - \frac {x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3} - \frac {\Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {1}{4} \\ \frac {5}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 x \Gamma \left (\frac {5}{6}\right )} - \frac {2 {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{3}} - \frac {\Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, \frac {1}{4} \\ - \frac {1}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{7} \Gamma \left (- \frac {1}{6}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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