Optimal. Leaf size=26 \[ \frac {2 \sqrt [4]{x^6+1} \left (x^6-5 x^4+1\right )}{5 x^5} \]
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Rubi [A] time = 0.10, antiderivative size = 31, normalized size of antiderivative = 1.19, 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 )^{5/4}}{5 x^5}-\frac {2 \sqrt [4]{x^6+1}}{x} \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^6 \left (1+x^6\right )^{3/4}} \, dx &=\int \left (\frac {2 x^3-x^9}{x^5 \left (1+x^6\right )^{3/4}}+\frac {-2-x^6+x^{12}}{x^6 \left (1+x^6\right )^{3/4}}\right ) \, dx\\ &=\int \frac {2 x^3-x^9}{x^5 \left (1+x^6\right )^{3/4}} \, dx+\int \frac {-2-x^6+x^{12}}{x^6 \left (1+x^6\right )^{3/4}} \, dx\\ &=\int \frac {2-x^6}{x^2 \left (1+x^6\right )^{3/4}} \, dx+\int \frac {\left (-2+x^6\right ) \sqrt [4]{1+x^6}}{x^6} \, dx\\ &=-\frac {2 \sqrt [4]{1+x^6}}{x}+\frac {2 \left (1+x^6\right )^{5/4}}{5 x^5}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 105, normalized size = 4.04 \begin {gather*} -x \, _2F_1\left (\frac {1}{6},\frac {3}{4};\frac {7}{6};-x^6\right )-\frac {2 \, _2F_1\left (-\frac {1}{6},\frac {3}{4};\frac {5}{6};-x^6\right )}{x}+\frac {1}{7} x^7 \, _2F_1\left (\frac {3}{4},\frac {7}{6};\frac {13}{6};-x^6\right )-\frac {1}{5} x^5 \, _2F_1\left (\frac {3}{4},\frac {5}{6};\frac {11}{6};-x^6\right )+\frac {2 \, _2F_1\left (-\frac {5}{6},\frac {3}{4};\frac {1}{6};-x^6\right )}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.21, size = 26, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{1+x^6} \left (1-5 x^4+x^6\right )}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 22, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (x^{6} - 5 \, x^{4} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}}{5 \, 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 {{\left (x^{6} - x^{4} + 1\right )} {\left (x^{6} - 2\right )}}{{\left (x^{6} + 1\right )}^{\frac {3}{4}} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 23, normalized size = 0.88
method | result | size |
trager | \(\frac {2 \left (x^{6}+1\right )^{\frac {1}{4}} \left (x^{6}-5 x^{4}+1\right )}{5 x^{5}}\) | \(23\) |
risch | \(\frac {\frac {2}{5} x^{12}+\frac {4}{5} x^{6}+\frac {2}{5}-2 x^{10}-2 x^{4}}{\left (x^{6}+1\right )^{\frac {3}{4}} x^{5}}\) | \(33\) |
gosper | \(\frac {2 \left (x^{4}-x^{2}+1\right ) \left (x^{2}+1\right ) \left (x^{6}-5 x^{4}+1\right )}{5 x^{5} \left (x^{6}+1\right )^{\frac {3}{4}}}\) | \(38\) |
meijerg | \(\frac {\hypergeom \left (\left [\frac {3}{4}, \frac {7}{6}\right ], \left [\frac {13}{6}\right ], -x^{6}\right ) x^{7}}{7}-\frac {\hypergeom \left (\left [\frac {3}{4}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], -x^{6}\right ) x^{5}}{5}-\hypergeom \left (\left [\frac {1}{6}, \frac {3}{4}\right ], \left [\frac {7}{6}\right ], -x^{6}\right ) x +\frac {2 \hypergeom \left (\left [-\frac {5}{6}, \frac {3}{4}\right ], \left [\frac {1}{6}\right ], -x^{6}\right )}{5 x^{5}}-\frac {2 \hypergeom \left (\left [-\frac {1}{6}, \frac {3}{4}\right ], \left [\frac {5}{6}\right ], -x^{6}\right )}{x}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 44, normalized size = 1.69 \begin {gather*} \frac {2 \, {\left (x^{12} - 5 \, x^{10} + 2 \, x^{6} - 5 \, x^{4} + 1\right )}}{5 \, {\left (x^{4} - x^{2} + 1\right )}^{\frac {3}{4}} {\left (x^{2} + 1\right )}^{\frac {3}{4}} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 27, normalized size = 1.04 \begin {gather*} \frac {2\,{\left (x^6+1\right )}^{5/4}-10\,x^4\,{\left (x^6+1\right )}^{1/4}}{5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.45, size = 155, normalized size = 5.96 \begin {gather*} \frac {x^{7} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {13}{6}\right )} - \frac {x^{5} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{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 \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{6}, \frac {3}{4} \\ \frac {7}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {7}{6}\right )} + \frac {\Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {3}{4} \\ \frac {5}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x \Gamma \left (\frac {5}{6}\right )} - \frac {\Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, \frac {3}{4} \\ \frac {1}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{5} \Gamma \left (\frac {1}{6}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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