Integrand size = 13, antiderivative size = 16 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\frac {\left (-1+x^3\right )^{4/3}}{4 x^4} \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\frac {\left (x^3-1\right )^{4/3}}{4 x^4} \]
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Rule 270
Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^3\right )^{4/3}}{4 x^4} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\frac {\left (-1+x^3\right )^{4/3}}{4 x^4} \]
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Time = 0.91 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(13\) |
pseudoelliptic | \(\frac {\left (x^{3}-1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(13\) |
gosper | \(\frac {\left (x -1\right ) \left (x^{2}+x +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{4 x^{4}}\) | \(22\) |
risch | \(\frac {x^{6}-2 x^{3}+1}{4 \left (x^{3}-1\right )^{\frac {2}{3}} x^{4}}\) | \(23\) |
meijerg | \(-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {4}{3}}}{4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x^{4}}\) | \(33\) |
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none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \]
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Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 129, normalized size of antiderivative = 8.06 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.17 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \]
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\[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{5}} \,d x } \]
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Time = 5.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx=-\frac {{\left (x^3-1\right )}^{1/3}-x^3\,{\left (x^3-1\right )}^{1/3}}{4\,x^4} \]
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