Integrand size = 37, antiderivative size = 120 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\frac {3 \left (-1+4 x^2\right ) \sqrt [3]{-x+x^3}}{4 x^3}-\frac {1}{8} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-9 \log (x)+9 \log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right )+4 \log (x) \text {$\#$1}^3-4 \log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \]
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Result contains complex when optimal does not.
Time = 1.02 (sec) , antiderivative size = 725, normalized size of antiderivative = 6.04, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {2081, 6860, 270, 477, 476, 486, 597, 12, 503} \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=-\frac {\sqrt [6]{3} \left (-2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \arctan \left (\frac {1+\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{\sqrt {3}-i} \sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{\left (\sqrt {3}-i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\sqrt [6]{3} \left (2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \arctan \left (\frac {1+\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{\sqrt {3}+i} \sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{\left (\sqrt {3}+i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {3 \left (2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x}}{4 \left (\sqrt {3}+i\right ) x}+\frac {3 \left (-2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x}}{4 \left (-\sqrt {3}+i\right ) x}-\frac {\left (\sqrt {3}+6 i\right ) \sqrt [3]{x^3-x}}{4 \left (-\sqrt {3}+i\right ) x^3}-\frac {\left (-\sqrt {3}+6 i\right ) \sqrt [3]{x^3-x}}{4 \left (\sqrt {3}+i\right ) x^3}-\frac {3 \sqrt [3]{x^3-x} \left (1-x^2\right )}{8 x^3}-\frac {\left (2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \log \left (-x^2-i \sqrt {3}+1\right )}{2 \sqrt [3]{3} \left (\sqrt {3}+i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {\left (-2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \log \left (-x^2+i \sqrt {3}+1\right )}{2 \sqrt [3]{3} \left (\sqrt {3}-i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}-\frac {3^{2/3} \left (-2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{\sqrt {3}-i} \sqrt [3]{x^2-1}\right )}{2 \left (\sqrt {3}-i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}}+\frac {3^{2/3} \left (2 \sqrt {3}+i\right ) \sqrt [3]{x^3-x} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{\sqrt {3}+i} \sqrt [3]{x^2-1}\right )}{2 \left (\sqrt {3}+i\right )^{7/3} \sqrt [3]{x} \sqrt [3]{x^2-1}} \]
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Rule 12
Rule 270
Rule 476
Rule 477
Rule 486
Rule 503
Rule 597
Rule 2081
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{-1+x^2} \left (8-10 x^2+x^4\right )}{x^{11/3} \left (4-2 x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = \frac {\sqrt [3]{-x+x^3} \int \left (\frac {\sqrt [3]{-1+x^2}}{x^{11/3}}+\frac {4 \left (1-2 x^2\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (4-2 x^2+x^4\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (4 \sqrt [3]{-x+x^3}\right ) \int \frac {\left (1-2 x^2\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (4-2 x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (4 \sqrt [3]{-x+x^3}\right ) \int \left (\frac {\left (-2+\frac {i}{\sqrt {3}}\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (-2-2 i \sqrt {3}+2 x^2\right )}+\frac {\left (-2-\frac {i}{\sqrt {3}}\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (-2+2 i \sqrt {3}+2 x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (4 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3} \left (-2-2 i \sqrt {3}+2 x^2\right )} \, dx}{3 \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (4 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3} \left (-2+2 i \sqrt {3}+2 x^2\right )} \, dx}{3 \sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (4 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^6}}{x^9 \left (-2-2 i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (4 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^6}}{x^9 \left (-2+2 i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (2 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{x^5 \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (2 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{x^5 \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {\left (6 i-\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x^3}-\frac {\left (6 i+\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x^3}-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}-\frac {\left (\left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {2 \left (3-i \sqrt {3}\right )-6 x^3}{x^2 \left (-1+x^3\right )^{2/3} \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \left (1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (\left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {2 \left (3+i \sqrt {3}\right )-6 x^3}{x^2 \left (-1+x^3\right )^{2/3} \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \left (1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {\left (6 i-\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x^3}-\frac {\left (6 i+\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x^3}+\frac {3 \left (i-2 \sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x}+\frac {3 \left (i+2 \sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x}-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (\left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {16 i \sqrt {3} x}{\left (-1+x^3\right )^{2/3} \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{8 \left (1+i \sqrt {3}\right )^2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (\left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int -\frac {16 i \sqrt {3} x}{\left (-1+x^3\right )^{2/3} \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{8 \left (1-i \sqrt {3}\right )^2 \sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {\left (6 i-\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x^3}-\frac {\left (6 i+\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x^3}+\frac {3 \left (i-2 \sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x}+\frac {3 \left (i+2 \sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x}-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (2 i \sqrt {3} \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3} \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\left (1+i \sqrt {3}\right )^2 \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (2 i \sqrt {3} \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3} \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\left (1-i \sqrt {3}\right )^2 \sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ & = -\frac {\left (6 i-\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x^3}-\frac {\left (6 i+\sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x^3}+\frac {3 \left (i-2 \sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i-\sqrt {3}\right ) x}+\frac {3 \left (i+2 \sqrt {3}\right ) \sqrt [3]{-x+x^3}}{4 \left (i+\sqrt {3}\right ) x}-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}-\frac {\sqrt [6]{3} \left (i-2 \sqrt {3}\right ) \sqrt [3]{-x+x^3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{-i+\sqrt {3}} \sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{\left (-i+\sqrt {3}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [6]{3} \left (i+2 \sqrt {3}\right ) \sqrt [3]{-x+x^3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{3} x^{2/3}}{\sqrt [3]{i+\sqrt {3}} \sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{\left (i+\sqrt {3}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (i+2 \sqrt {3}\right ) \sqrt [3]{-x+x^3} \log \left (1-i \sqrt {3}-x^2\right )}{2 \sqrt [3]{3} \left (i+\sqrt {3}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (i-2 \sqrt {3}\right ) \sqrt [3]{-x+x^3} \log \left (1+i \sqrt {3}-x^2\right )}{2 \sqrt [3]{3} \left (-i+\sqrt {3}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {3^{2/3} \left (i-2 \sqrt {3}\right ) \sqrt [3]{-x+x^3} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{-i+\sqrt {3}} \sqrt [3]{-1+x^2}\right )}{2 \left (-i+\sqrt {3}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {3^{2/3} \left (i+2 \sqrt {3}\right ) \sqrt [3]{-x+x^3} \log \left (\sqrt [6]{3} x^{2/3}-\sqrt [3]{i+\sqrt {3}} \sqrt [3]{-1+x^2}\right )}{2 \left (i+\sqrt {3}\right )^{7/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=-\frac {\sqrt [3]{x \left (-1+x^2\right )} \left (18 \left (1-4 x^2\right ) \sqrt [3]{-1+x^2}+x^{8/3} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-18 \log (x)+27 \log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right )+8 \log (x) \text {$\#$1}^3-12 \log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ]\right )}{24 x^3 \sqrt [3]{-1+x^2}} \]
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Time = 42.73 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(\frac {\left (24 x^{2}-6\right ) \left (x^{3}-x \right )^{\frac {1}{3}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{6}-6 \textit {\_Z}^{3}+3\right )}{\sum }\frac {\left (4 \textit {\_R}^{3}-9\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (4 \textit {\_R}^{3}-3\right )}\right ) x^{3}}{8 x^{3}}\) | \(83\) |
trager | \(\text {Expression too large to display}\) | \(2117\) |
risch | \(\text {Expression too large to display}\) | \(3327\) |
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Exception generated. \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 2.89 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x^{4} - 10 x^{2} + 8\right )}{x^{4} \left (x^{4} - 2 x^{2} + 4\right )}\, dx \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.31 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 10 \, x^{2} + 8\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{{\left (x^{4} - 2 \, x^{2} + 4\right )} x^{4}} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.31 \[ \int \frac {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx=\int \frac {{\left (x^3-x\right )}^{1/3}\,\left (x^4-10\,x^2+8\right )}{x^4\,\left (x^4-2\,x^2+4\right )} \,d x \]
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