Optimal. Leaf size=31 \[ \frac {2 \sqrt [4]{x^6-1} \left (x^{12}+9 x^8-2 x^6+1\right )}{9 x^9} \]
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Rubi [A] time = 0.11, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1833, 1584, 449, 1586, 1478} \begin {gather*} \frac {2 \sqrt [4]{x^6-1}}{x}+\frac {2 \left (x^6-1\right )^{9/4}}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 1478
Rule 1584
Rule 1586
Rule 1833
Rubi steps
\begin {align*} \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^{10} \left (-1+x^6\right )^{3/4}} \, dx &=\int \left (\frac {2 x^6+x^{12}}{x^8 \left (-1+x^6\right )^{3/4}}+\frac {2-3 x^6+x^{18}}{x^{10} \left (-1+x^6\right )^{3/4}}\right ) \, dx\\ &=\int \frac {2 x^6+x^{12}}{x^8 \left (-1+x^6\right )^{3/4}} \, dx+\int \frac {2-3 x^6+x^{18}}{x^{10} \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 {\sqrt [4]{-1+x^6} \left (-2+x^6+x^{12}\right )}{x^{10}} \, dx\\ &=\frac {2 \sqrt [4]{-1+x^6}}{x}+\int \frac {\left (-1+x^6\right )^{5/4} \left (2+x^6\right )}{x^{10}} \, dx\\ &=\frac {2 \sqrt [4]{-1+x^6}}{x}+\frac {2 \left (-1+x^6\right )^{9/4}}{9 x^9}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 168, normalized size = 5.42 \begin {gather*} \frac {45 \left (1-x^6\right )^{3/4} x^6 \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {1}{2};x^6\right )-10 \left (1-x^6\right )^{3/4} \, _2F_1\left (-\frac {3}{2},\frac {3}{4};-\frac {1}{2};x^6\right )-90 \left (1-x^6\right )^{3/4} x^8 \, _2F_1\left (-\frac {1}{6},\frac {3}{4};\frac {5}{6};x^6\right )+x^{12} \left (10 \left (1-x^6\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};x^6\right )+9 \left (1-x^6\right )^{3/4} x^2 \, _2F_1\left (\frac {3}{4},\frac {5}{6};\frac {11}{6};x^6\right )+10 \left (x^6-1\right )\right )}{45 x^9 \left (x^6-1\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 8.91, size = 31, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{x^6-1} \left (x^{12}+9 x^8-2 x^6+1\right )}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 27, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (x^{12} + 9 \, x^{8} - 2 \, x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}}}{9 \, x^{9}} \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^{12} + x^{8} - 2 \, x^{6} + 1\right )} {\left (x^{6} + 2\right )}}{{\left (x^{6} - 1\right )}^{\frac {3}{4}} x^{10}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 1.55 \begin {gather*} \frac {2 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{12}+9 x^{8}-2 x^{6}+1\right )}{9 x^{9} \left (x^{6}-1\right )^{\frac {3}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.90, size = 58, normalized size = 1.87 \begin {gather*} \frac {2 \, {\left (x^{18} + 9 \, x^{14} - 3 \, x^{12} - 9 \, x^{8} + 3 \, x^{6} - 1\right )}}{9 \, {\left (x^{2} + x + 1\right )}^{\frac {3}{4}} {\left (x^{2} - x + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}^{\frac {3}{4}} x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 49, normalized size = 1.58 \begin {gather*} \frac {2\,{\left (x^6-1\right )}^{1/4}}{x}-\frac {4\,{\left (x^6-1\right )}^{1/4}}{9\,x^3}+\frac {2\,x^3\,{\left (x^6-1\right )}^{1/4}}{9}+\frac {2\,{\left (x^6-1\right )}^{1/4}}{9\,x^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.02, size = 139, normalized size = 4.48 \begin {gather*} \frac {x^{9} e^{- \frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {x^{6}} \right )}}{9} + \frac {x^{5} e^{- \frac {3 i \pi }{4}} \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}} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} - \frac {e^{\frac {i \pi }{4}} \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}} \right )}}{3 x \Gamma \left (\frac {5}{6}\right )} - \frac {e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{6}} \right )}}{x^{3}} + \frac {2 e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ - \frac {1}{2} \end {matrix}\middle | {x^{6}} \right )}}{9 x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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