Integrand size = 29, antiderivative size = 110 \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {\left (-2-3 x^3\right ) \left (1-x^3\right )^{2/3}}{10 x^5}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1-x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1-x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right ) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {21, 485, 597, 12, 384} \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{6} \log \left (2 x^3-1\right )-\frac {1}{2} \log \left (x-\sqrt [3]{1-x^3}\right )-\frac {\left (1-x^3\right )^{2/3}}{5 x^5}-\frac {3 \left (1-x^3\right )^{2/3}}{10 x^2} \]
[In]
[Out]
Rule 12
Rule 21
Rule 384
Rule 485
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\left (1-x^3\right )^{5/3}}{x^6 \left (-1+2 x^3\right )} \, dx \\ & = -\frac {\left (1-x^3\right )^{2/3}}{5 x^5}+\frac {1}{5} \int \frac {-3+x^3}{x^3 \sqrt [3]{1-x^3} \left (-1+2 x^3\right )} \, dx \\ & = -\frac {\left (1-x^3\right )^{2/3}}{5 x^5}-\frac {3 \left (1-x^3\right )^{2/3}}{10 x^2}+\frac {1}{10} \int -\frac {10}{\sqrt [3]{1-x^3} \left (-1+2 x^3\right )} \, dx \\ & = -\frac {\left (1-x^3\right )^{2/3}}{5 x^5}-\frac {3 \left (1-x^3\right )^{2/3}}{10 x^2}-\int \frac {1}{\sqrt [3]{1-x^3} \left (-1+2 x^3\right )} \, dx \\ & = -\frac {\left (1-x^3\right )^{2/3}}{5 x^5}-\frac {3 \left (1-x^3\right )^{2/3}}{10 x^2}+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{6} \log \left (-1+2 x^3\right )-\frac {1}{2} \log \left (x-\sqrt [3]{1-x^3}\right ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=-\frac {\left (1-x^3\right )^{2/3} \left (2+3 x^3\right )}{10 x^5}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1-x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1-x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right ) \]
[In]
[Out]
Time = 2.64 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {-10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (-x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}+5 \ln \left (\frac {x^{2}+x \left (-x^{3}+1\right )^{\frac {1}{3}}+\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-10 \ln \left (\frac {-x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}-9 \left (-x^{3}+1\right )^{\frac {2}{3}} x^{3}-6 \left (-x^{3}+1\right )^{\frac {2}{3}}}{30 x^{5}}\) | \(119\) |
trager | \(-\frac {\left (3 x^{3}+2\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{10 x^{5}}-\frac {\ln \left (-\frac {290589696 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}-10252800 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x +50677344 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-23337024 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}-421089 \left (-x^{3}+1\right )^{\frac {2}{3}} x -106800 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-73232 x^{3}-2324717568 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}+14158560 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+64078}{2 x^{3}-1}\right )}{3}+32 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \ln \left (-\frac {84363264 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}+10252800 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x +40424544 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+9178464 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}-527889 \left (-x^{3}+1\right )^{\frac {2}{3}} x +106800 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+220717 x^{3}-674906112 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-23337024 \operatorname {RootOf}\left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+31531}{2 x^{3}-1}\right )\) | \(348\) |
risch | \(\frac {3 x^{6}-x^{3}-2}{10 x^{5} \left (-x^{3}+1\right )^{\frac {1}{3}}}-\frac {\ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-2 \left (-x^{3}+1\right )^{\frac {2}{3}} x +\left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-x^{3}+1}{2 x^{3}-1}\right )}{3}+\frac {\ln \left (-\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+\left (-x^{3}+1\right )^{\frac {2}{3}} x -2 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+1}{2 x^{3}-1}\right )}{3}-\ln \left (-\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+\left (-x^{3}+1\right )^{\frac {2}{3}} x -2 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+1}{2 x^{3}-1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )\) | \(420\) |
[In]
[Out]
none
Time = 0.53 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.24 \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {4 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (x^{3} - 1\right )}}{7 \, x^{3} + 1}\right ) - 5 \, x^{5} \log \left (\frac {2 \, x^{3} - 3 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - 1}{2 \, x^{3} - 1}\right ) - 3 \, {\left (3 \, x^{3} + 2\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \]
[In]
[Out]
\[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x - 1\right ) \left (x^{2} + x + 1\right )}{x^{6} \cdot \left (2 x^{3} - 1\right )}\, dx \]
[In]
[Out]
\[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx=-\int \frac {{\left (1-x^3\right )}^{5/3}}{x^6\,\left (2\,x^3-1\right )} \,d x \]
[In]
[Out]