Integrand size = 18, antiderivative size = 33 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^{11}} \, dx=\frac {\sqrt [3]{1+x^3} \left (7-9 x^3-4 x^6+12 x^9\right )}{70 x^{10}} \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {464, 277, 270} \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^{11}} \, dx=\frac {\left (x^3+1\right )^{4/3}}{10 x^{10}}-\frac {8 \left (x^3+1\right )^{4/3}}{35 x^7}+\frac {6 \left (x^3+1\right )^{4/3}}{35 x^4} \]
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Rule 270
Rule 277
Rule 464
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+x^3\right )^{4/3}}{10 x^{10}}+\frac {8}{5} \int \frac {\sqrt [3]{1+x^3}}{x^8} \, dx \\ & = \frac {\left (1+x^3\right )^{4/3}}{10 x^{10}}-\frac {8 \left (1+x^3\right )^{4/3}}{35 x^7}-\frac {24}{35} \int \frac {\sqrt [3]{1+x^3}}{x^5} \, dx \\ & = \frac {\left (1+x^3\right )^{4/3}}{10 x^{10}}-\frac {8 \left (1+x^3\right )^{4/3}}{35 x^7}+\frac {6 \left (1+x^3\right )^{4/3}}{35 x^4} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^{11}} \, dx=\frac {\left (1+x^3\right )^{4/3} \left (7-16 x^3+12 x^6\right )}{70 x^{10}} \]
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Time = 0.87 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {\left (12 x^{6}-16 x^{3}+7\right ) \left (x^{3}+1\right )^{\frac {4}{3}}}{70 x^{10}}\) | \(25\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {1}{3}} \left (12 x^{9}-4 x^{6}-9 x^{3}+7\right )}{70 x^{10}}\) | \(30\) |
risch | \(\frac {12 x^{12}+8 x^{9}-13 x^{6}-2 x^{3}+7}{70 \left (x^{3}+1\right )^{\frac {2}{3}} x^{10}}\) | \(35\) |
gosper | \(\frac {\left (x^{2}-x +1\right ) \left (1+x \right ) \left (12 x^{6}-16 x^{3}+7\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{70 x^{10}}\) | \(36\) |
meijerg | \(\frac {\left (\frac {9}{14} x^{9}-\frac {3}{14} x^{6}+\frac {1}{7} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{10 x^{10}}-\frac {\left (-\frac {3}{4} x^{6}+\frac {1}{4} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{7 x^{7}}\) | \(55\) |
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^{11}} \, dx=\frac {{\left (12 \, x^{9} - 4 \, x^{6} - 9 \, x^{3} + 7\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{70 \, x^{10}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (29) = 58\).
Time = 1.34 (sec) , antiderivative size = 199, normalized size of antiderivative = 6.03 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^{11}} \, dx=- \frac {2 \sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{x^{3} + 1} \Gamma \left (- \frac {7}{3}\right )}{3 x \Gamma \left (- \frac {1}{3}\right )} + \frac {2 \sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{x^{3} + 1} \Gamma \left (- \frac {7}{3}\right )}{9 x^{4} \Gamma \left (- \frac {1}{3}\right )} - \frac {4 \sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{27 x^{6} \Gamma \left (- \frac {1}{3}\right )} - \frac {4 \sqrt [3]{x^{3} + 1} \Gamma \left (- \frac {7}{3}\right )}{9 x^{7} \Gamma \left (- \frac {1}{3}\right )} - \frac {28 \sqrt [3]{1 + \frac {1}{x^{3}}} \Gamma \left (- \frac {10}{3}\right )}{27 x^{9} \Gamma \left (- \frac {1}{3}\right )} \]
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Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^{11}} \, dx=\frac {{\left (x^{3} + 1\right )}^{\frac {4}{3}}}{2 \, x^{4}} - \frac {3 \, {\left (x^{3} + 1\right )}^{\frac {7}{3}}}{7 \, x^{7}} + \frac {{\left (x^{3} + 1\right )}^{\frac {10}{3}}}{10 \, x^{10}} \]
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\[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^{11}} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{11}} \,d x } \]
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Time = 5.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^{11}} \, dx=\frac {6\,{\left (x^3+1\right )}^{1/3}}{35\,x}-\frac {2\,{\left (x^3+1\right )}^{1/3}}{35\,x^4}-\frac {9\,{\left (x^3+1\right )}^{1/3}}{70\,x^7}+\frac {{\left (x^3+1\right )}^{1/3}}{10\,x^{10}} \]
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