Integrand size = 13, antiderivative size = 16 \[ \int \frac {\sqrt [3]{-1+x^6}}{x^9} \, dx=\frac {\left (-1+x^6\right )^{4/3}}{8 x^8} \]
[Out]
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} \[ \int \frac {\sqrt [3]{-1+x^6}}{x^9} \, dx=\frac {\left (x^6-1\right )^{4/3}}{8 x^8} \]
[In]
[Out]
Rule 270
Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^6\right )^{4/3}}{8 x^8} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^6}}{x^9} \, dx=\frac {\left (-1+x^6\right )^{4/3}}{8 x^8} \]
[In]
[Out]
Time = 0.82 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(\frac {\left (x^{6}-1\right )^{\frac {4}{3}}}{8 x^{8}}\) | \(13\) |
pseudoelliptic | \(\frac {\left (x^{6}-1\right )^{\frac {4}{3}}}{8 x^{8}}\) | \(13\) |
risch | \(\frac {x^{12}-2 x^{6}+1}{8 \left (x^{6}-1\right )^{\frac {2}{3}} x^{8}}\) | \(23\) |
gosper | \(\frac {\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{8 x^{8}}\) | \(33\) |
meijerg | \(-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \left (-x^{6}+1\right )^{\frac {4}{3}}}{8 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{8}}\) | \(33\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{-1+x^6}}{x^9} \, dx=\frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 129, normalized size of antiderivative = 8.06 \[ \int \frac {\sqrt [3]{-1+x^6}}{x^9} \, dx=\begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{-1+x^6}}{x^9} \, dx=\frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} \]
[In]
[Out]
\[ \int \frac {\sqrt [3]{-1+x^6}}{x^9} \, dx=\int { \frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{9}} \,d x } \]
[In]
[Out]
Time = 5.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{-1+x^6}}{x^9} \, dx=\frac {{\left (x^6-1\right )}^{4/3}}{8\,x^8} \]
[In]
[Out]