Integrand size = 13, antiderivative size = 93 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx=-\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
[Out]
Time = 0.00 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {283, 245} \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx=\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {\left (x^3-1\right )^{2/3}}{2 x^2} \]
[In]
[Out]
Rule 245
Rule 283
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}+\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx=\frac {1}{6} \left (-\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )-2 \log \left (-x+\sqrt [3]{-1+x^3}\right )+\log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.35
method | result | size |
meijerg | \(-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], x^{3}\right )}{2 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2}}\) | \(33\) |
risch | \(-\frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(43\) |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}-2 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}+\ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}-3 \left (x^{3}-1\right )^{\frac {2}{3}}}{6 x^{2}}\) | \(94\) |
trager | \(-\frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\frac {\ln \left (182271728 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{3}-775851456 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -775851456 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-821419388 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1303197526 x^{3}-1458173824 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}-1129625404 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )+849911430\right )}{3}-\frac {4 \ln \left (182271728 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{3}-775851456 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -775851456 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-821419388 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-1303197526 x^{3}-1458173824 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}-1129625404 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )+849911430\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )}{3}+\frac {4 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \ln \left (6512 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{3}+38784 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +38784 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+37156 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{3}+6963 x \left (x^{3}-1\right )^{\frac {2}{3}}+6963 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+7370 x^{3}-52096 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}-22116 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-2345\right )}{3}\) | \(450\) |
[In]
[Out]
none
Time = 0.44 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.13 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) - x^{2} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) - 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.69 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.40 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx=\frac {e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
[In]
[Out]
\[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}}{x^3} \,d x \]
[In]
[Out]