Integrand size = 32, antiderivative size = 116 \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=-\frac {3 \left (-2+7 x^2\right ) \sqrt [3]{-2 x+x^3}}{16 x^3}+\frac {1}{16} \text {RootSum}\left [2-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-6 \log (x)+6 \log \left (\sqrt [3]{-2 x+x^3}-x \text {$\#$1}\right )+11 \log (x) \text {$\#$1}^3-11 \log \left (\sqrt [3]{-2 x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(561\) vs. \(2(116)=232\).
Time = 0.61 (sec) , antiderivative size = 561, normalized size of antiderivative = 4.84, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2081, 6860, 477, 476, 486, 597, 12, 503} \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\frac {\sqrt {3} \sqrt [3]{\frac {1}{2} \left (674 \sqrt {2}-953\right )} \sqrt [3]{x^3-2 x} \arctan \left (\frac {\frac {2 \sqrt [3]{2-\sqrt {2}} x^{2/3}}{\sqrt [3]{x^2-2}}+1}{\sqrt {3}}\right )}{16 \sqrt [3]{x} \sqrt [3]{x^2-2}}-\frac {\sqrt {3} \sqrt [3]{\frac {1}{2} \left (953+674 \sqrt {2}\right )} \sqrt [3]{x^3-2 x} \arctan \left (\frac {\frac {2 \sqrt [3]{2+\sqrt {2}} x^{2/3}}{\sqrt [3]{x^2-2}}+1}{\sqrt {3}}\right )}{16 \sqrt [3]{x} \sqrt [3]{x^2-2}}-\frac {21 \left (4+3 \sqrt {2}\right ) \sqrt [3]{x^3-2 x}}{128 x}-\frac {21 \left (4-3 \sqrt {2}\right ) \sqrt [3]{x^3-2 x}}{128 x}+\frac {3 \left (4+\sqrt {2}\right ) \sqrt [3]{x^3-2 x}}{64 x^3}+\frac {3 \left (4-\sqrt {2}\right ) \sqrt [3]{x^3-2 x}}{64 x^3}-\frac {\sqrt [3]{\frac {1}{2} \left (674 \sqrt {2}-953\right )} \sqrt [3]{x^3-2 x} \log \left (2 \left (1+\sqrt {2}\right )-x^2\right )}{32 \sqrt [3]{x} \sqrt [3]{x^2-2}}+\frac {\sqrt [3]{\frac {1}{2} \left (953+674 \sqrt {2}\right )} \sqrt [3]{x^3-2 x} \log \left (x^2-2 \left (1-\sqrt {2}\right )\right )}{32 \sqrt [3]{x} \sqrt [3]{x^2-2}}+\frac {3 \sqrt [3]{\frac {1}{2} \left (674 \sqrt {2}-953\right )} \sqrt [3]{x^3-2 x} \log \left (\sqrt [3]{2-\sqrt {2}} x^{2/3}-\sqrt [3]{x^2-2}\right )}{32 \sqrt [3]{x} \sqrt [3]{x^2-2}}-\frac {3 \sqrt [3]{\frac {1}{2} \left (953+674 \sqrt {2}\right )} \sqrt [3]{x^3-2 x} \log \left (\sqrt [3]{2+\sqrt {2}} x^{2/3}-\sqrt [3]{x^2-2}\right )}{32 \sqrt [3]{x} \sqrt [3]{x^2-2}} \]
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Rule 12
Rule 476
Rule 477
Rule 486
Rule 503
Rule 597
Rule 2081
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{-2 x+x^3} \int \frac {\sqrt [3]{-2+x^2} \left (4+x^2\right )}{x^{11/3} \left (-4-4 x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+x^2}} \\ & = \frac {\sqrt [3]{-2 x+x^3} \int \left (\frac {\left (1+\frac {3}{\sqrt {2}}\right ) \sqrt [3]{-2+x^2}}{x^{11/3} \left (-4-4 \sqrt {2}+2 x^2\right )}+\frac {\left (1-\frac {3}{\sqrt {2}}\right ) \sqrt [3]{-2+x^2}}{x^{11/3} \left (-4+4 \sqrt {2}+2 x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-2+x^2}} \\ & = \frac {\left (\left (2-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \int \frac {\sqrt [3]{-2+x^2}}{x^{11/3} \left (-4+4 \sqrt {2}+2 x^2\right )} \, dx}{2 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\left (\left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \int \frac {\sqrt [3]{-2+x^2}}{x^{11/3} \left (-4-4 \sqrt {2}+2 x^2\right )} \, dx}{2 \sqrt [3]{x} \sqrt [3]{-2+x^2}} \\ & = \frac {\left (3 \left (2-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-2+x^6}}{x^9 \left (-4+4 \sqrt {2}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\left (3 \left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-2+x^6}}{x^9 \left (-4-4 \sqrt {2}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{-2+x^2}} \\ & = \frac {\left (3 \left (2-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-2+x^3}}{x^5 \left (-4+4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\left (3 \left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-2+x^3}}{x^5 \left (-4-4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x} \sqrt [3]{-2+x^2}} \\ & = \frac {3 \left (4-\sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{64 x^3}+\frac {3 \left (4+\sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{64 x^3}+\frac {\left (3 \left (2-3 \sqrt {2}\right ) \left (1+\sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {4 \left (3+\sqrt {2}\right )-6 x^3}{x^2 \left (-2+x^3\right )^{2/3} \left (-4+4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{64 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\left (3 \left (1-\sqrt {2}\right ) \left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {4 \left (3-\sqrt {2}\right )-6 x^3}{x^2 \left (-2+x^3\right )^{2/3} \left (-4-4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{64 \sqrt [3]{x} \sqrt [3]{-2+x^2}} \\ & = \frac {3 \left (4-\sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{64 x^3}+\frac {3 \left (4+\sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{64 x^3}-\frac {21 \left (4-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{128 x}-\frac {21 \left (4+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{128 x}+\frac {\left (3 \left (2-3 \sqrt {2}\right ) \left (1+\sqrt {2}\right )^2 \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int -\frac {64 \sqrt {2} x}{\left (-2+x^3\right )^{2/3} \left (-4+4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{512 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\left (3 \left (1-\sqrt {2}\right )^2 \left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {64 \sqrt {2} x}{\left (-2+x^3\right )^{2/3} \left (-4-4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{512 \sqrt [3]{x} \sqrt [3]{-2+x^2}} \\ & = \frac {3 \left (4-\sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{64 x^3}+\frac {3 \left (4+\sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{64 x^3}-\frac {21 \left (4-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{128 x}-\frac {21 \left (4+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{128 x}-\frac {\left (3 \left (2-3 \sqrt {2}\right ) \left (1+\sqrt {2}\right )^2 \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (-2+x^3\right )^{2/3} \left (-4+4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt {2} \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\left (3 \left (1-\sqrt {2}\right )^2 \left (2+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (-2+x^3\right )^{2/3} \left (-4-4 \sqrt {2}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt {2} \sqrt [3]{x} \sqrt [3]{-2+x^2}} \\ & = \frac {3 \left (4-\sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{64 x^3}+\frac {3 \left (4+\sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{64 x^3}-\frac {21 \left (4-3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{128 x}-\frac {21 \left (4+3 \sqrt {2}\right ) \sqrt [3]{-2 x+x^3}}{128 x}+\frac {\sqrt {3} \sqrt [3]{\frac {1}{2} \left (-953+674 \sqrt {2}\right )} \sqrt [3]{-2 x+x^3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2-\sqrt {2}} x^{2/3}}{\sqrt [3]{-2+x^2}}}{\sqrt {3}}\right )}{16 \sqrt [3]{x} \sqrt [3]{-2+x^2}}-\frac {\sqrt {3} \sqrt [3]{\frac {1}{2} \left (953+674 \sqrt {2}\right )} \sqrt [3]{-2 x+x^3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2+\sqrt {2}} x^{2/3}}{\sqrt [3]{-2+x^2}}}{\sqrt {3}}\right )}{16 \sqrt [3]{x} \sqrt [3]{-2+x^2}}-\frac {\sqrt [3]{\frac {1}{2} \left (-953+674 \sqrt {2}\right )} \sqrt [3]{-2 x+x^3} \log \left (2 \left (1+\sqrt {2}\right )-x^2\right )}{32 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {\sqrt [3]{\frac {1}{2} \left (953+674 \sqrt {2}\right )} \sqrt [3]{-2 x+x^3} \log \left (-2 \left (1-\sqrt {2}\right )+x^2\right )}{32 \sqrt [3]{x} \sqrt [3]{-2+x^2}}+\frac {3 \sqrt [3]{\frac {1}{2} \left (-953+674 \sqrt {2}\right )} \sqrt [3]{-2 x+x^3} \log \left (\sqrt [3]{2-\sqrt {2}} x^{2/3}-\sqrt [3]{-2+x^2}\right )}{32 \sqrt [3]{x} \sqrt [3]{-2+x^2}}-\frac {3 \sqrt [3]{\frac {1}{2} \left (953+674 \sqrt {2}\right )} \sqrt [3]{-2 x+x^3} \log \left (\sqrt [3]{2+\sqrt {2}} x^{2/3}-\sqrt [3]{-2+x^2}\right )}{32 \sqrt [3]{x} \sqrt [3]{-2+x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.22 \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\frac {\sqrt [3]{x \left (-2+x^2\right )} \left (9 \left (2-7 x^2\right ) \sqrt [3]{-2+x^2}+x^{8/3} \text {RootSum}\left [2-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-12 \log (x)+18 \log \left (\sqrt [3]{-2+x^2}-x^{2/3} \text {$\#$1}\right )+22 \log (x) \text {$\#$1}^3-33 \log \left (\sqrt [3]{-2+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ]\right )}{48 x^3 \sqrt [3]{-2+x^2}} \]
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Time = 185.73 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-4 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\left (11 \textit {\_R}^{3}-6\right ) \ln \left (\frac {-\textit {\_R} x +{\left (x \left (x^{2}-2\right )\right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (\textit {\_R}^{3}-2\right )}\right ) x^{3}-21 {\left (x \left (x^{2}-2\right )\right )}^{\frac {1}{3}} x^{2}+6 {\left (x \left (x^{2}-2\right )\right )}^{\frac {1}{3}}}{16 x^{3}}\) | \(88\) |
trager | \(\text {Expression too large to display}\) | \(10597\) |
risch | \(\text {Expression too large to display}\) | \(14280\) |
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Exception generated. \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 51.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\int \frac {\sqrt [3]{x \left (x^{2} - 2\right )} \left (x^{2} + 4\right )}{x^{4} \left (x^{4} - 4 x^{2} - 4\right )}\, dx \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.20 \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{3} - 2 \, x\right )}^{\frac {1}{3}} {\left (x^{2} + 4\right )}}{{\left (x^{4} - 4 \, x^{2} - 4\right )} x^{4}} \,d x } \]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.28 \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{3} - 2 \, x\right )}^{\frac {1}{3}} {\left (x^{2} + 4\right )}}{{\left (x^{4} - 4 \, x^{2} - 4\right )} x^{4}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.30 \[ \int \frac {\left (4+x^2\right ) \sqrt [3]{-2 x+x^3}}{x^4 \left (-4-4 x^2+x^4\right )} \, dx=\int -\frac {\left (x^2+4\right )\,{\left (x^3-2\,x\right )}^{1/3}}{x^4\,\left (-x^4+4\,x^2+4\right )} \,d x \]
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