Integrand size = 30, antiderivative size = 107 \[ \int \frac {\left (-2+x^2\right ) \sqrt [3]{x+x^3}}{x^2 \left (4-2 x^2+x^4\right )} \, dx=\frac {3 \sqrt [3]{x+x^3}}{4 x}-\frac {1}{8} \text {RootSum}\left [7-10 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-7 \log (x)+7 \log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )+4 \log (x) \text {$\#$1}^3-4 \log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \]
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Result contains complex when optimal does not.
Time = 0.75 (sec) , antiderivative size = 763, normalized size of antiderivative = 7.13, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2081, 6860, 477, 476, 486, 12, 503} \[ \int \frac {\left (-2+x^2\right ) \sqrt [3]{x+x^3}}{x^2 \left (4-2 x^2+x^4\right )} \, dx=-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{x^3+x} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}} \sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{4 \left (-\sqrt {3}+i\right ) \sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}} \sqrt [3]{x} \sqrt [3]{x^2+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{x^3+x} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+2 i}} \sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{4 \left (\sqrt {3}+i\right ) \sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+2 i}} \sqrt [3]{x} \sqrt [3]{x^2+1}}+\frac {\left (\sqrt {3}+3 i\right ) \sqrt [3]{x^3+x}}{4 \left (\sqrt {3}+i\right ) x}+\frac {\left (-\sqrt {3}+3 i\right ) \sqrt [3]{x^3+x}}{4 \left (-\sqrt {3}+i\right ) x}-\frac {\left (\sqrt {3}+3 i\right ) \sqrt [3]{x^3+x} \log \left (-x^2-i \sqrt {3}+1\right )}{24 \left (\sqrt {3}+i\right ) \sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+2 i}} \sqrt [3]{x} \sqrt [3]{x^2+1}}-\frac {\left (-\sqrt {3}+3 i\right ) \sqrt [3]{x^3+x} \log \left (-x^2+i \sqrt {3}+1\right )}{24 \left (-\sqrt {3}+i\right ) \sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}} \sqrt [3]{x} \sqrt [3]{x^2+1}}+\frac {\left (-\sqrt {3}+3 i\right ) \sqrt [3]{x^3+x} \log \left (x^{2/3}-\sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}} \sqrt [3]{x^2+1}\right )}{8 \left (-\sqrt {3}+i\right ) \sqrt [3]{\frac {-\sqrt {3}+i}{-\sqrt {3}+2 i}} \sqrt [3]{x} \sqrt [3]{x^2+1}}+\frac {\left (\sqrt {3}+3 i\right ) \sqrt [3]{x^3+x} \log \left (x^{2/3}-\sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+2 i}} \sqrt [3]{x^2+1}\right )}{8 \left (\sqrt {3}+i\right ) \sqrt [3]{\frac {\sqrt {3}+i}{\sqrt {3}+2 i}} \sqrt [3]{x} \sqrt [3]{x^2+1}} \]
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Rule 12
Rule 476
Rule 477
Rule 486
Rule 503
Rule 2081
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{x+x^3} \int \frac {\left (-2+x^2\right ) \sqrt [3]{1+x^2}}{x^{5/3} \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 {\left (1+\frac {i}{\sqrt {3}}\right ) \sqrt [3]{1+x^2}}{x^{5/3} \left (-2-2 i \sqrt {3}+2 x^2\right )}+\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \sqrt [3]{1+x^2}}{x^{5/3} \left (-2+2 i \sqrt {3}+2 x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{1+x^2}}{x^{5/3} \left (-2+2 i \sqrt {3}+2 x^2\right )} \, dx}{3 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{1+x^2}}{x^{5/3} \left (-2-2 i \sqrt {3}+2 x^2\right )} \, dx}{3 \sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6}}{x^3 \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 (\left (3+i \sqrt {3}\right ) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^6}}{x^3 \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 (\left (3-i \sqrt {3}\right ) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3}}{x^2 \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (\left (3+i \sqrt {3}\right ) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1+x^3}}{x^2 \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {\left (3 i-\sqrt {3}\right ) \sqrt [3]{x+x^3}}{4 \left (i-\sqrt {3}\right ) x}+\frac {\left (3 i+\sqrt {3}\right ) \sqrt [3]{x+x^3}}{4 \left (i+\sqrt {3}\right ) x}-\frac {\left (\left (3-i \sqrt {3}\right ) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {\left (-4+2 i \sqrt {3}\right ) x}{\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 (3+i \sqrt {3}\right ) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {\left (-4-2 i \sqrt {3}\right ) x}{\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 (3 i-\sqrt {3}\right ) \sqrt [3]{x+x^3}}{4 \left (i-\sqrt {3}\right ) x}+\frac {\left (3 i+\sqrt {3}\right ) \sqrt [3]{x+x^3}}{4 \left (i+\sqrt {3}\right ) x}+\frac {\left (\left (2-i \sqrt {3}\right ) \left (3-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 )}{2 \left (1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (\left (2+i \sqrt {3}\right ) \left (3+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 )}{2 \left (1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+x^2}} \\ & = \frac {\left (3 i-\sqrt {3}\right ) \sqrt [3]{x+x^3}}{4 \left (i-\sqrt {3}\right ) x}+\frac {\left (3 i+\sqrt {3}\right ) \sqrt [3]{x+x^3}}{4 \left (i+\sqrt {3}\right ) x}-\frac {\left (5 i+\sqrt {3}\right ) \sqrt [3]{x+x^3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{\frac {i-\sqrt {3}}{2 i-\sqrt {3}}} \sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{4 \left (-i+\sqrt {3}\right )^{4/3} \left (-2 i+\sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{2 i+\sqrt {3}} \sqrt [3]{x+x^3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{\frac {i+\sqrt {3}}{2 i+\sqrt {3}}} \sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{4 \left (i+\sqrt {3}\right )^{4/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\sqrt [3]{2 i+\sqrt {3}} \left (3 i+\sqrt {3}\right ) \sqrt [3]{x+x^3} \log \left (1-i \sqrt {3}-x^2\right )}{24 \left (i+\sqrt {3}\right )^{4/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (3+5 i \sqrt {3}\right ) \sqrt [3]{x+x^3} \log \left (1+i \sqrt {3}-x^2\right )}{24 \left (-i+\sqrt {3}\right )^{4/3} \left (-2 i+\sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left (3+5 i \sqrt {3}\right ) \sqrt [3]{x+x^3} \log \left (x^{2/3}-\sqrt [3]{\frac {i-\sqrt {3}}{2 i-\sqrt {3}}} \sqrt [3]{1+x^2}\right )}{8 \left (-i+\sqrt {3}\right )^{4/3} \left (-2 i+\sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\sqrt [3]{2 i+\sqrt {3}} \left (3 i+\sqrt {3}\right ) \sqrt [3]{x+x^3} \log \left (x^{2/3}-\sqrt [3]{\frac {i+\sqrt {3}}{2 i+\sqrt {3}}} \sqrt [3]{1+x^2}\right )}{8 \left (i+\sqrt {3}\right )^{4/3} \sqrt [3]{x} \sqrt [3]{1+x^2}} \\ \end{align*}
Time = 1.79 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.31 \[ \int \frac {\left (-2+x^2\right ) \sqrt [3]{x+x^3}}{x^2 \left (4-2 x^2+x^4\right )} \, dx=\frac {\sqrt [3]{x+x^3} \left (6 \sqrt [3]{1+x^2}-\frac {1}{3} x^{2/3} \text {RootSum}\left [7-10 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-14 \log (x)+21 \log \left (\sqrt [3]{1+x^2}-x^{2/3} \text {$\#$1}\right )+8 \log (x) \text {$\#$1}^3-12 \log \left (\sqrt [3]{1+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ]\right )}{8 x \sqrt [3]{1+x^2}} \]
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Time = 215.35 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{6}-10 \textit {\_Z}^{3}+7\right )}{\sum }\frac {\left (4 \textit {\_R}^{3}-7\right ) \ln \left (\frac {-\textit {\_R} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (4 \textit {\_R}^{3}-5\right )}\right ) x +6 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{8 x}\) | \(75\) |
risch | \(\text {Expression too large to display}\) | \(3092\) |
trager | \(\text {Expression too large to display}\) | \(4774\) |
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Exception generated. \[ \int \frac {\left (-2+x^2\right ) \sqrt [3]{x+x^3}}{x^2 \left (4-2 x^2+x^4\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 2.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.27 \[ \int \frac {\left (-2+x^2\right ) \sqrt [3]{x+x^3}}{x^2 \left (4-2 x^2+x^4\right )} \, dx=\int \frac {\sqrt [3]{x \left (x^{2} + 1\right )} \left (x^{2} - 2\right )}{x^{2} \left (x^{4} - 2 x^{2} + 4\right )}\, dx \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.91 \[ \int \frac {\left (-2+x^2\right ) \sqrt [3]{x+x^3}}{x^2 \left (4-2 x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{2} - 2\right )}}{{\left (x^{4} - 2 \, x^{2} + 4\right )} x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {\left (-2+x^2\right ) \sqrt [3]{x+x^3}}{x^2 \left (4-2 x^2+x^4\right )} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 5.68 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.28 \[ \int \frac {\left (-2+x^2\right ) \sqrt [3]{x+x^3}}{x^2 \left (4-2 x^2+x^4\right )} \, dx=\int \frac {\left (x^2-2\right )\,{\left (x^3+x\right )}^{1/3}}{x^2\,\left (x^4-2\,x^2+4\right )} \,d x \]
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