Integrand size = 35, antiderivative size = 163 \[ \int \frac {\left (1+x^3\right ) \sqrt {-2-x^3+x^6}}{x^4 \left (-1-2 x^3+x^6\right )} \, dx=\frac {\sqrt {-2-x^3+x^6}}{3 x^3}-\frac {\arctan \left (\frac {\sqrt {-2-x^3+x^6}}{\sqrt {2} \left (1+x^3\right )}\right )}{\sqrt {2}}+\frac {1}{3} \sqrt {4+3 \sqrt {2}} \arctan \left (\frac {\sqrt {-4+3 \sqrt {2}} \sqrt {-2-x^3+x^6}}{1+x^3}\right )-\frac {1}{3} \sqrt {-4+3 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt {4+3 \sqrt {2}} \sqrt {-2-x^3+x^6}}{1+x^3}\right ) \]
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Time = 0.55 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.37, number of steps used = 26, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {6860, 1371, 746, 857, 635, 212, 738, 210, 748, 6847, 1033, 1090, 1046} \[ \int \frac {\left (1+x^3\right ) \sqrt {-2-x^3+x^6}}{x^4 \left (-1-2 x^3+x^6\right )} \, dx=\frac {1}{3} \sqrt {2} \arctan \left (\frac {x^3+4}{2 \sqrt {2} \sqrt {x^6-x^3-2}}\right )-\frac {\arctan \left (\frac {x^3+4}{2 \sqrt {2} \sqrt {x^6-x^3-2}}\right )}{6 \sqrt {2}}-\frac {\left (1+\sqrt {2}\right ) \arctan \left (\frac {-\left (\left (1-2 \sqrt {2}\right ) x^3\right )-\sqrt {2}+5}{2 \sqrt [4]{2} \sqrt {x^6-x^3-2}}\right )}{3\ 2^{3/4}}-\frac {\left (1-\sqrt {2}\right ) \text {arctanh}\left (\frac {-\left (\left (1+2 \sqrt {2}\right ) x^3\right )+\sqrt {2}+5}{2 \sqrt [4]{2} \sqrt {x^6-x^3-2}}\right )}{3\ 2^{3/4}}+\frac {\sqrt {x^6-x^3-2}}{3 x^3} \]
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Rule 210
Rule 212
Rule 635
Rule 738
Rule 746
Rule 748
Rule 857
Rule 1033
Rule 1046
Rule 1090
Rule 1371
Rule 6847
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {-2-x^3+x^6}}{x^4}+\frac {\sqrt {-2-x^3+x^6}}{x}-\frac {x^2 \left (-3+x^3\right ) \sqrt {-2-x^3+x^6}}{-1-2 x^3+x^6}\right ) \, dx \\ & = -\int \frac {\sqrt {-2-x^3+x^6}}{x^4} \, dx+\int \frac {\sqrt {-2-x^3+x^6}}{x} \, dx-\int \frac {x^2 \left (-3+x^3\right ) \sqrt {-2-x^3+x^6}}{-1-2 x^3+x^6} \, dx \\ & = -\left (\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {-2-x+x^2}}{x^2} \, dx,x,x^3\right )\right )+\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {-2-x+x^2}}{x} \, dx,x,x^3\right )-\frac {1}{3} \text {Subst}\left (\int \frac {(-3+x) \sqrt {-2-x+x^2}}{-1-2 x+x^2} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-2-x^3+x^6}}{3 x^3}-\frac {1}{6} \text {Subst}\left (\int \frac {4+x}{x \sqrt {-2-x+x^2}} \, dx,x,x^3\right )-\frac {1}{6} \text {Subst}\left (\int \frac {-1+2 x}{x \sqrt {-2-x+x^2}} \, dx,x,x^3\right )+\frac {1}{3} \text {Subst}\left (\int \frac {-\frac {11}{2}-x+\frac {3 x^2}{2}}{\left (-1-2 x+x^2\right ) \sqrt {-2-x+x^2}} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-2-x^3+x^6}}{3 x^3}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {-2-x+x^2}} \, dx,x,x^3\right )+\frac {1}{6} \text {Subst}\left (\int \frac {1}{x \sqrt {-2-x+x^2}} \, dx,x,x^3\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-2-x+x^2}} \, dx,x,x^3\right )+\frac {1}{3} \text {Subst}\left (\int \frac {-4+2 x}{\left (-1-2 x+x^2\right ) \sqrt {-2-x+x^2}} \, dx,x,x^3\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-2-x+x^2}} \, dx,x,x^3\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {-2-x+x^2}} \, dx,x,x^3\right ) \\ & = \frac {\sqrt {-2-x^3+x^6}}{3 x^3}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,\frac {-4-x^3}{\sqrt {-2-x^3+x^6}}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x^3}{\sqrt {-2-x^3+x^6}}\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x^3}{\sqrt {-2-x^3+x^6}}\right )+\frac {4}{3} \text {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,\frac {-4-x^3}{\sqrt {-2-x^3+x^6}}\right )+\frac {1}{3} \left (2-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\left (-2-2 \sqrt {2}+2 x\right ) \sqrt {-2-x+x^2}} \, dx,x,x^3\right )+\frac {1}{3} \left (2+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\left (-2+2 \sqrt {2}+2 x\right ) \sqrt {-2-x+x^2}} \, dx,x,x^3\right )+\text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x^3}{\sqrt {-2-x^3+x^6}}\right ) \\ & = \frac {\sqrt {-2-x^3+x^6}}{3 x^3}-\frac {\arctan \left (\frac {4+x^3}{2 \sqrt {2} \sqrt {-2-x^3+x^6}}\right )}{6 \sqrt {2}}+\frac {1}{3} \sqrt {2} \arctan \left (\frac {4+x^3}{2 \sqrt {2} \sqrt {-2-x^3+x^6}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {1-2 x^3}{2 \sqrt {-2-x^3+x^6}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {-1+2 x^3}{2 \sqrt {-2-x^3+x^6}}\right )-\frac {1}{3} \left (2 \left (2-\sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-32+8 \left (-2-2 \sqrt {2}\right )+4 \left (-2-2 \sqrt {2}\right )^2-x^2} \, dx,x,-\frac {2 \left (5+\sqrt {2}+\left (1-2 \left (1+\sqrt {2}\right )\right ) x^3\right )}{\sqrt {-2-x^3+x^6}}\right )-\frac {1}{3} \left (2 \left (2+\sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-32+8 \left (-2+2 \sqrt {2}\right )+4 \left (-2+2 \sqrt {2}\right )^2-x^2} \, dx,x,\frac {2 \left (-5+\sqrt {2}+\left (-1-2 \left (-1+\sqrt {2}\right )\right ) x^3\right )}{\sqrt {-2-x^3+x^6}}\right ) \\ & = \frac {\sqrt {-2-x^3+x^6}}{3 x^3}-\frac {\arctan \left (\frac {4+x^3}{2 \sqrt {2} \sqrt {-2-x^3+x^6}}\right )}{6 \sqrt {2}}+\frac {1}{3} \sqrt {2} \arctan \left (\frac {4+x^3}{2 \sqrt {2} \sqrt {-2-x^3+x^6}}\right )-\frac {\left (1+\sqrt {2}\right ) \arctan \left (\frac {5-\sqrt {2}-\left (1-2 \sqrt {2}\right ) x^3}{2 \sqrt [4]{2} \sqrt {-2-x^3+x^6}}\right )}{3\ 2^{3/4}}+\frac {1}{2} \text {arctanh}\left (\frac {1-2 x^3}{2 \sqrt {-2-x^3+x^6}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {-1+2 x^3}{2 \sqrt {-2-x^3+x^6}}\right )-\frac {\left (1-\sqrt {2}\right ) \text {arctanh}\left (\frac {5+\sqrt {2}-\left (1+2 \sqrt {2}\right ) x^3}{2 \sqrt [4]{2} \sqrt {-2-x^3+x^6}}\right )}{3\ 2^{3/4}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.99 \[ \int \frac {\left (1+x^3\right ) \sqrt {-2-x^3+x^6}}{x^4 \left (-1-2 x^3+x^6\right )} \, dx=\frac {\sqrt {-2-x^3+x^6}}{3 x^3}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {-2-x^3+x^6}}{-2+x^3}\right )}{\sqrt {2}}-\frac {1}{3} \sqrt {4+3 \sqrt {2}} \arctan \left (\frac {\sqrt {2+\frac {3}{\sqrt {2}}} \sqrt {-2-x^3+x^6}}{-2+x^3}\right )-\frac {1}{3} \sqrt {-4+3 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt {-2+\frac {3}{\sqrt {2}}} \sqrt {-2-x^3+x^6}}{-2+x^3}\right ) \]
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Time = 7.35 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {-2 x^{3} \left (2^{\frac {1}{4}}+2^{\frac {3}{4}}\right ) \arctan \left (\frac {\left (2 \sqrt {2}\, x^{3}-x^{3}+5-\sqrt {2}\right ) 2^{\frac {3}{4}}}{4 \sqrt {x^{6}-x^{3}-2}}\right )+2 x^{3} \left (2^{\frac {1}{4}}-2^{\frac {3}{4}}\right ) \operatorname {arctanh}\left (\frac {\left (2 \sqrt {2}\, x^{3}+x^{3}-\sqrt {2}-5\right ) 2^{\frac {3}{4}}}{4 \sqrt {x^{6}-x^{3}-2}}\right )+3 \sqrt {2}\, \arctan \left (\frac {\left (x^{3}+4\right ) \sqrt {2}}{4 \sqrt {x^{6}-x^{3}-2}}\right ) x^{3}+4 \sqrt {x^{6}-x^{3}-2}}{12 x^{3}}\) | \(152\) |
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Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (130) = 260\).
Time = 0.25 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.67 \[ \int \frac {\left (1+x^3\right ) \sqrt {-2-x^3+x^6}}{x^4 \left (-1-2 x^3+x^6\right )} \, dx=-\frac {3 \, \sqrt {2} x^{3} \arctan \left (-\frac {1}{2} \, \sqrt {2} x^{3} + \frac {1}{2} \, \sqrt {2} \sqrt {x^{6} - x^{3} - 2}\right ) + x^{3} \sqrt {3 \, \sqrt {2} - 4} \log \left (-x^{3} + \sqrt {3 \, \sqrt {2} - 4} {\left (\sqrt {2} + 1\right )} + \sqrt {2} + \sqrt {x^{6} - x^{3} - 2} + 1\right ) - x^{3} \sqrt {3 \, \sqrt {2} - 4} \log \left (-x^{3} - \sqrt {3 \, \sqrt {2} - 4} {\left (\sqrt {2} + 1\right )} + \sqrt {2} + \sqrt {x^{6} - x^{3} - 2} + 1\right ) - x^{3} \sqrt {-3 \, \sqrt {2} - 4} \log \left (-x^{3} + {\left (\sqrt {2} - 1\right )} \sqrt {-3 \, \sqrt {2} - 4} - \sqrt {2} + \sqrt {x^{6} - x^{3} - 2} + 1\right ) + x^{3} \sqrt {-3 \, \sqrt {2} - 4} \log \left (-x^{3} - {\left (\sqrt {2} - 1\right )} \sqrt {-3 \, \sqrt {2} - 4} - \sqrt {2} + \sqrt {x^{6} - x^{3} - 2} + 1\right ) - 2 \, x^{3} - 2 \, \sqrt {x^{6} - x^{3} - 2}}{6 \, x^{3}} \]
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Timed out. \[ \int \frac {\left (1+x^3\right ) \sqrt {-2-x^3+x^6}}{x^4 \left (-1-2 x^3+x^6\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (1+x^3\right ) \sqrt {-2-x^3+x^6}}{x^4 \left (-1-2 x^3+x^6\right )} \, dx=\int { \frac {\sqrt {x^{6} - x^{3} - 2} {\left (x^{3} + 1\right )}}{{\left (x^{6} - 2 \, x^{3} - 1\right )} x^{4}} \,d x } \]
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Exception generated. \[ \int \frac {\left (1+x^3\right ) \sqrt {-2-x^3+x^6}}{x^4 \left (-1-2 x^3+x^6\right )} \, dx=\text {Exception raised: AttributeError} \]
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Timed out. \[ \int \frac {\left (1+x^3\right ) \sqrt {-2-x^3+x^6}}{x^4 \left (-1-2 x^3+x^6\right )} \, dx=\int -\frac {\left (x^3+1\right )\,\sqrt {x^6-x^3-2}}{x^4\,\left (-x^6+2\,x^3+1\right )} \,d x \]
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