Integrand size = 18, antiderivative size = 28 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^8} \, dx=\frac {\sqrt [3]{-1+x^3} \left (-2-3 x^3+5 x^6\right )}{14 x^7} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {464, 270} \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^8} \, dx=\frac {\left (x^3-1\right )^{4/3}}{7 x^7}+\frac {5 \left (x^3-1\right )^{4/3}}{14 x^4} \]
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Rule 270
Rule 464
Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^3\right )^{4/3}}{7 x^7}+\frac {10}{7} \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx \\ & = \frac {\left (-1+x^3\right )^{4/3}}{7 x^7}+\frac {5 \left (-1+x^3\right )^{4/3}}{14 x^4} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^8} \, dx=\frac {\sqrt [3]{-1+x^3} \left (-2-3 x^3+5 x^6\right )}{14 x^7} \]
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Time = 0.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(\frac {\left (x^{3}-1\right )^{\frac {4}{3}} \left (5 x^{3}+2\right )}{14 x^{7}}\) | \(20\) |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (5 x^{6}-3 x^{3}-2\right )}{14 x^{7}}\) | \(25\) |
risch | \(\frac {5 x^{9}-8 x^{6}+x^{3}+2}{14 \left (x^{3}-1\right )^{\frac {2}{3}} x^{7}}\) | \(28\) |
gosper | \(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (5 x^{3}+2\right ) \left (x -1\right ) \left (x^{2}+x +1\right )}{14 x^{7}}\) | \(29\) |
meijerg | \(-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {4}{3}}}{4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x^{4}}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {3}{4} x^{6}-\frac {1}{4} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}{7 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x^{7}}\) | \(78\) |
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^8} \, dx=\frac {{\left (5 \, x^{6} - 3 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{14 \, x^{7}} \]
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Result contains complex when optimal does not.
Time = 1.13 (sec) , antiderivative size = 423, normalized size of antiderivative = 15.11 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^8} \, dx=\begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} + \begin {cases} \frac {3 x^{6} \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {2 x^{3} \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} + \frac {4 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{9} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{6} \Gamma \left (- \frac {1}{3}\right )} - \frac {5 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {4 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^8} \, dx=\frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{2 \, x^{4}} - \frac {{\left (x^{3} - 1\right )}^{\frac {7}{3}}}{7 \, x^{7}} \]
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\[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^8} \, dx=\int { \frac {{\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{8}} \,d x } \]
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Time = 5.46 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^8} \, dx=-\frac {2\,{\left (x^3-1\right )}^{1/3}+3\,x^3\,{\left (x^3-1\right )}^{1/3}-5\,x^6\,{\left (x^3-1\right )}^{1/3}}{14\,x^7} \]
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