Integrand size = 29, antiderivative size = 114 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=-\frac {3 \sqrt [3]{-x+2 x^3}}{2 x}+\frac {1}{4} \text {RootSum}\left [5-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{-x+2 x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-x+2 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ] \]
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Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 437, normalized size of antiderivative = 3.83, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2081, 6857, 477, 476, 486, 12, 503} \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt [3]{2-i} \sqrt {3} \sqrt [3]{2 x^3-x} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2-i} x^{2/3}}{\sqrt [3]{2 x^2-1}}}{\sqrt {3}}\right )}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt [3]{2+i} \sqrt {3} \sqrt [3]{2 x^3-x} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2+i} x^{2/3}}{\sqrt [3]{2 x^2-1}}}{\sqrt {3}}\right )}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}-\frac {3 \sqrt [3]{2 x^3-x}}{2 x}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \sqrt [3]{2+i} \sqrt [3]{2 x^3-x} \log \left (-x^2+i\right )}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \sqrt [3]{2-i} \sqrt [3]{2 x^3-x} \log \left (x^2+i\right )}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}-\frac {\left (\frac {3}{8}-\frac {3 i}{8}\right ) \sqrt [3]{2-i} \sqrt [3]{2 x^3-x} \log \left (-\sqrt [3]{2 x^2-1}+\sqrt [3]{2-i} x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}-\frac {\left (\frac {3}{8}+\frac {3 i}{8}\right ) \sqrt [3]{2+i} \sqrt [3]{2 x^3-x} \log \left (-\sqrt [3]{2 x^2-1}+\sqrt [3]{2+i} x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}} \]
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Rule 12
Rule 476
Rule 477
Rule 486
Rule 503
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{-x+2 x^3} \int \frac {\left (1+x^2\right ) \sqrt [3]{-1+2 x^2}}{x^{5/3} \left (1+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}} \\ & = \frac {\sqrt [3]{-x+2 x^3} \int \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [3]{-1+2 x^2}}{x^{5/3} \left (i-x^2\right )}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [3]{-1+2 x^2}}{x^{5/3} \left (i+x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}} \\ & = -\frac {\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [3]{-x+2 x^3}\right ) \int \frac {\sqrt [3]{-1+2 x^2}}{x^{5/3} \left (i-x^2\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}+\frac {\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [3]{-x+2 x^3}\right ) \int \frac {\sqrt [3]{-1+2 x^2}}{x^{5/3} \left (i+x^2\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}} \\ & = -\frac {\left (\left (\frac {3}{2}-\frac {3 i}{2}\right ) \sqrt [3]{-x+2 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+2 x^6}}{x^3 \left (i-x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}+\frac {\left (\left (\frac {3}{2}+\frac {3 i}{2}\right ) \sqrt [3]{-x+2 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+2 x^6}}{x^3 \left (i+x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}} \\ & = -\frac {\left (\left (\frac {3}{4}-\frac {3 i}{4}\right ) \sqrt [3]{-x+2 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+2 x^3}}{x^2 \left (i-x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}+\frac {\left (\left (\frac {3}{4}+\frac {3 i}{4}\right ) \sqrt [3]{-x+2 x^3}\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-1+2 x^3}}{x^2 \left (i+x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}} \\ & = -\frac {3 \sqrt [3]{-x+2 x^3}}{2 x}+\frac {\left (\left (\frac {3}{4}-\frac {3 i}{4}\right ) \sqrt [3]{-x+2 x^3}\right ) \text {Subst}\left (\int \frac {(1+2 i) x}{\left (i+x^3\right ) \left (-1+2 x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}+\frac {\left (\left (\frac {3}{4}+\frac {3 i}{4}\right ) \sqrt [3]{-x+2 x^3}\right ) \text {Subst}\left (\int -\frac {(1-2 i) x}{\left (i-x^3\right ) \left (-1+2 x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}} \\ & = -\frac {3 \sqrt [3]{-x+2 x^3}}{2 x}+-\frac {\left (\left (\frac {9}{4}-\frac {3 i}{4}\right ) \sqrt [3]{-x+2 x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (i-x^3\right ) \left (-1+2 x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}+\frac {\left (\left (\frac {9}{4}+\frac {3 i}{4}\right ) \sqrt [3]{-x+2 x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (i+x^3\right ) \left (-1+2 x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}} \\ & = -\frac {3 \sqrt [3]{-x+2 x^3}}{2 x}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt [3]{2-i} \sqrt {3} \sqrt [3]{-x+2 x^3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2-i} x^{2/3}}{\sqrt [3]{-1+2 x^2}}}{\sqrt {3}}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt [3]{2+i} \sqrt {3} \sqrt [3]{-x+2 x^3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2+i} x^{2/3}}{\sqrt [3]{-1+2 x^2}}}{\sqrt {3}}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \sqrt [3]{2+i} \sqrt [3]{-x+2 x^3} \log \left (i-x^2\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \sqrt [3]{2-i} \sqrt [3]{-x+2 x^3} \log \left (i+x^2\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}-\frac {\left (\frac {3}{8}-\frac {3 i}{8}\right ) \sqrt [3]{2-i} \sqrt [3]{-x+2 x^3} \log \left (\sqrt [3]{2-i} x^{2/3}-\sqrt [3]{-1+2 x^2}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}}-\frac {\left (\frac {3}{8}+\frac {3 i}{8}\right ) \sqrt [3]{2+i} \sqrt [3]{-x+2 x^3} \log \left (\sqrt [3]{2+i} x^{2/3}-\sqrt [3]{-1+2 x^2}\right )}{\sqrt [3]{x} \sqrt [3]{-1+2 x^2}} \\ \end{align*}
Time = 1.78 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.18 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=\frac {18-36 x^2+x^{2/3} \left (-1+2 x^2\right )^{2/3} \text {RootSum}\left [5-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-10 \log (x)+15 \log \left (\sqrt [3]{-1+2 x^2}-x^{2/3} \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-3 \log \left (\sqrt [3]{-1+2 x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ]}{12 \left (x \left (-1+2 x^2\right )\right )^{2/3}} \]
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Time = 168.42 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(\frac {-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-4 \textit {\_Z}^{3}+5\right )}{\sum }\frac {\left (\textit {\_R}^{3}-5\right ) \ln \left (\frac {-\textit {\_R} x +\left (2 x^{3}-x \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (\textit {\_R}^{3}-2\right )}\right ) x -6 \left (2 x^{3}-x \right )^{\frac {1}{3}}}{4 x}\) | \(74\) |
trager | \(\text {Expression too large to display}\) | \(10722\) |
risch | \(\text {Expression too large to display}\) | \(11848\) |
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Exception generated. \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 1.71 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=\int \frac {\sqrt [3]{x \left (2 x^{2} - 1\right )} \left (x^{2} + 1\right )}{x^{2} \left (x^{4} + 1\right )}\, dx \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{{\left (x^{4} + 1\right )} x^{2}} \,d x } \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{{\left (x^{4} + 1\right )} x^{2}} \,d x } \]
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Not integrable
Time = 5.65 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=\int \frac {{\left (2\,x^3-x\right )}^{1/3}\,\left (x^2+1\right )}{x^2\,\left (x^4+1\right )} \,d x \]
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